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Algebraic Aspects in Tropical Mathematics

Much like in the theory of algebraic geometry, we develop a correspondence between certain types of algebraic and geometric objects. The basic algebraic environment we work in is the a semifield of fractions H(x1,...,xn) of the polynomial semidomain H[x1,...,xn], where H is taken to be an idempotent semifield, while for the geometric environment we have the space H^n. We show that taking H to be idempotent makes both H(x1,...,xn) and Hn idempotent which turn out to satisfy many desired properties that we utilize for our construction. The fundamental algebraic and geometric objects having interrelations are called kernels, encapsulating congruences over semifields and skeletons which serve as the analog for zero-sets of algebraic geometry. As an analog for the celebrated Nullstellenzats theorem, we develop a correspondence between skeletons and a family of kernels called polars originally developed in the theory of lattice-ordered groups. For a special kind of skeletons, called principal skeletons, we have simplified the correspondence by restricting our algebraic environment to a certain kernel of H(x1,...,xn). After establishing the linkage between kernels and skeletons we proceed to construct a second linkage, this time between a family of skeletons and what we call"corner-loci". Essentially a corner locus is what is called a tropical variety in the theory of tropical geometry, which is a set of corner roots of some set of tropical polynomials. A skeleton and a corner-locus define the exact same subset of H^n in different ways: while a corner locus is defined by corner roots of tropical polynomials, the skeleton is defined by equating fractions from H(x1,...,xn) to 1. All the connections presented above form a path connecting a tropical variety to a certain kind of kernel.

The fundamental algebraic and geometric objects having interrelations are called kernels, encapsulating congruences over semifields (analogous to ideals in algebraic geometry) and skeletons which serve as the analogue for zero-sets of algebraic geometry. As an analogue for the celebrated Nullstellenzats theorem which provides a correspondence between radical ideals and zero sets, we develop a correspondence between skeletons and a family of kernels called polars originally developed in the theory of lattice-ordered groups. For a special kind of skeletons, called principal skeletons, we have simplified the correspondence by restricting our algebraic environment to a very special semifield which is also a kernel of H(x 1 , ..., x n ).
After establishing the linkage between kernels and skeletons we proceed to construct a second linkage, this time between a family of skeletons and what we call 'corner-loci'. Essentially a corner locus is what is called a tropical variety in the theory of tropical geometry, which is a set of corner roots of some set of tropical polynomials. The relation between a skeleton and a corner-locus is that they define the exact same subset of H n , though in different ways: while a corner locus is defined by corner roots of tropical polynomials, the skeleton is defined by an equality, namely, equating fractions from H(x 1 , ..., x n ) to 1. All the connections presented above form a path connecting a tropical variety to a kernel. In this paper we also develop a correspondence between supertropical varieties (generalizing tropical varieties) introduced by Izhakian, Knebusch and Rowen to the lattice of principal kernels. Restricting this correspondence to a sublattice of kernels, whom we call regular kernels, yields the correspondence described above.
The research we conduct shows that the theory of supertropical geometry is in fact a natural generalization of tropical geometry.
We conclude by developing some algebraic structure notions such as composition series and convexity degree, along with some notions holding a geometric interpretation, like reducibility and hyperdimension.

Overview
In the following overview, we will establish a linkage between our construction and the widely known theory of algebraic geometry, in order to give the reader some additional insights.
As noted in the abstract, we develop a geometric structure corresponding to a semifield of fractions H(x 1 , ..., x n ) over a bipotent semifield H. We concentrate on the case where H is a divisible archimedean semifield, motivated by our interest to link the theory to tropical geometry. We also consider a divisible bipotent archimedean semifield R, which is also complete (and thus isomorphic to the positive reals with max-plus operations).
The role of ideals is played by an algebraic structures called kernels. While ideals encapsulate the structure of the preimages of zero of rings homomorphisms, kernels encapsulate the preimages of {1} with respect to semifield homomorphisms with the substantial difference that considering homomorphisms one gets a sublattice of kernels with respect to intersection and multiplication, (Con(R(x 1 , ..., x n )), ·, ∩).
For each kernel K of Con (R(x 1 , ..., x n )), we define an analogue for zero sets corresponding to ideals, namely, a skeleton Skel(K) ⊆ R n , which is defined to be the set of all points in R n over which all the elements of the kernel K are evaluated to be 1.
While in the classical theory all ideals are finitely generated, in our case we explicitly consider the finitely generated kernels in Con (R(x 1 , ..., x n )). This family of finitely generated kernels in Con (R(x 1 , ..., x n )) forms a sublattice of (Con(R(x 1 , ..., x n )), ·, ∩), which we denote as PCon (R(x 1 , ..., x n )). At this point, the theory is considerably simplified, since a well-known result in the theory of semifields states that every finitely generated kernel is principal, i.e., generated as a kernel by a single element.
The generator of a principal kernel is not unique, though there is a designated set of generators which provides us with many tools for implementing the theory.
First we establish a Zariski-like correspondence between a special kind of ker-nels called polars and general skeletons. Then we find some special semifield of R(x 1 , ..., x n ) over which the above correspondence gives rise to a simplified correspondence between principal (finitely generated) kernels of and principal (finitely generated) skeletons which hold most of our interest. We show that a skeleton is uniquely defined by any of its corresponding kernel generators, define a maximal kernel, and show that the maximal principal kernels correspond to points in R n .
Along the way, we provide some theory concerning general (not necessarily principal) kernels and skeleton.
The geometric interpretation established for the theory of semifields provides us with a topology, called the Stone Topology, defined on the family of the socalled irreducible kernels (analogous to prime ideals) and the family of maximal kernels. This topology is in essence the semifields version of the famous Zariski topology.
After establishing this geometric framework, we introduce a map linking supertropical varieties (cf. [4]), which we call 'Corner loci', to a subfamily of principal skeletons. We use the latter to construct a correspondence between principal supertropical varieties (the analogue for hypersurfaces of algebraic geometry) and a lattice of kernels generated by special principal kernels which we call cornerintegral. In addition, we characterize a family of skeletons that coincide with the family of 'regular' tropical varieties via which we obtain a correspondence with a sublattice of principal kernels called regular kernels.
Finally, we establish a notion of reducibility and some other notions and properties of kernels and their corresponding skeletons, such as convex-dependence, dimensionality etc. We begin our thesis by introducing the relevant results in the theory of semifields. A poset (X, ≤) is a lattice or lattice-ordered set if for a, b ∈ X, the set {a, b} has a join a ∨ b (also known as the least upper bound, or the supremum) and a meet a ∧ b (also known as the greatest lower bound, or the infimum). Equivalently, a lattice can be defined as a directed poset (X, ∨, ∧), consisting of a set X and two associative and commutative binary operations ∨ and ∧ defined on X, such that for all elements a, b ∈ X. a ∨ (a ∧ b) = a ∧ (a ∨ b) = a.
The first definition is derived from the second by defining a partial order on X by Definition 2.1.2. A lattice (X, ∨, ∧) is said to be distributive if the following condition holds for any a, b, c ∈ X: A fundamental result in lattice theory states that this condition is equivalent to the condition a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).
Definition 2.1.3. A conditionally complete lattice is a poset P in which every nonempty subset that has an upper bound in P has a least upper bound (i.e., a supremum) in P and every nonempty subset that has an lower bound in P has an infimum in P . A lattice P is said to be complete if all its subsets have both a join and a meet. Example 2.1.4. The poset of the real numbers R is conditionally complete, and become complete when adjoining −∞, ∞. Likewise the positive reals R + is conditionally complete, and become complete when adjoining 0, ∞. Note that for example the set {1/n : n ∈ N} ⊂ R + is not bounded below in R + since 0 ∈ R + . Definition 2.1.5. The subset A of the poset P is said to be completely closed in P if A contains the least upper bound or greatest lower bound of any of its nonempty subsets, if either exists in P .
Definition 2.1.6. Given two lattices (X, ∨ X , ∧ X ) and (Y, ∨ Y , ∧ Y ), a homomorphism of lattices or lattice homomorphism is a function f : X → Y such that

Semifields
Basic setting and assumptions Definition 2.2.1. A semiring S is a set S equipped with two binary operations + and ·, such that (S, +) is a commutative monoid with identity element 0, (S, ·) is a monoid with identity element 1, multiplication left and right distributes over addition and multiplication by 0 annihilates S. S is called commutative when (S, ·) is commutative. S is called a domain when S is multiplicatively cancellative.

Definition 2.2.2.
A semifield is a semiring (H, +, ·) in which all nonzero elements have a multiplicative inverse. A semifield is said to be proper if it is not a field. Note 2.2.3. Though in general, a semifield is not assumed to be commutative, in the scope of our study we consider commutative semifields. Thus, we always assume a semifield to be commutative. Nevertheless, we introduce some definitions and results in the wider context in which a semifield is not necessarily commutative. For example, the notion of a semifield-kernel is defined as a normal subgroup implying the definition refers to the wider context.
From now on, unless stated otherwise we assume a semifield to be commutative and proper.
Remark 2.2.7. As stated in Example 2.2.6(4) (additively) idempotent semifields correspond to lattice ordered groups. In the stated correspondence, the addition operation + of the semifield coincides with the so-called 'join' operation ∨ defined on the underlying lattice structure of the lattice ordered group. Due to this and in order to emphasize the underlying lattice structure of the semifield, when considering idempotent semifields we denote addition by ∔. Definition 2.2.8. An idempotent semiring is a semiring S such that ∀a ∈ S : a + a = a.
An idempotent semifield is an idempotent semiring which is a semifield. Note that considering a semifield the condition of idempotency is equivalent to demanding that 1 + 1 = 1, as a = 1a = (1 + 1)a = a + a.
Definition 2.2.9. We define a bipotent semiring S to be a semiring with 1 (multiplicative identity element) admitting bipotent addition, i.e., α + β ∈ {α, β} for any α, β ∈ S. When S is a semifield, i.e., every nonzero element of S is invertible with respect to multiplication, we say that S is a bipotent semifield. Remark 2.2.10. A bipotent semifield (semiring) is a special case of an idempotent semifield (semiring).
Motivated by tropical geometry, we have a special interest in the following particular semiring. Definition 2.2.11. Let (H, ·, 1) be a lattice ordered monoid. Define addition on H to be the operation of supremum, denoted by ∔, i.e., for any a, b ∈ H a ∔ b = sup(a, b). (2.1) Adjoin a zero element 0 to H such that ∀a ∈ H : a ∔ 0 = 0 ∔ a = a. Then H = (H, ·, ∔, 1, 0) is a semiring. If H is totally ordered then supremum is maximum and the semiring is bipotent. Taking H to be a multiplicative group, H becomes an idempotent semifield. In such a case, for a, b ∈ H, (a ∔ b) −1 = inf(a −1 , b −1 ). Thus ∔ induces a infimum operation, to be denoted by ∧, such that In particular, for any a ∈ H, (a ∔ a −1 ) −1 = a ∧ a −1 .
If H is totally ordered infimum is minimum.
Remark 2.2.12. The semifield described in Definition 2.2.11,is idempotent (bipotent) and thus can be viewed as a commutative lattice ordered-group (totally ordered) (H, ∔, ·) where for a, b ∈ H, a ≤ b ⇔ a ∔ b = b.
The following lemma and the subsequent remarks establish the connection between a semifield addition and a natural order it induces. Lemma 2.2.13. For every proper semifield H, the following quasi-identity holds: for a, b, c ∈ H.
Proof. Let a + b + c = a. Multiplying both sides of the equation by a −1 ba −1 and then adding ca −1 to both parts yield that (ba −1 + ca −1 )(1 + ba −1 ) = ba −1 + ca −1 . Hence, as multiplicative cancellation holds, we get 1 + ba −1 = 1 and therefore Remark 2.2.14. Every (commutative with respect to addition) semifield H is endowed with a partial order defined by and is ordered with respect to a natural order, i.e., a ≤ b implies that a+c ≤ b+c, ac ≤ bc and ca ≤ cb for all a, b, c ∈ H. In the special case of idempotent semifields, the relation (2.3) can be rephrased as

4)
Note 2.2.15. In the literature (for example in [3]), semifields are sometimes not assumed to be commutative with respect to addition and thus do not always have a natural order. Since the semifields we consider are additively commutative (abelian), a semifield in our scope is always partially ordered with respect to the natural order. Although x + 1 ≥ x and x = x + 1, one has x + 1 > x (for example take x = 1).
Indeed, the idempotency of H implies that M is idempotent with respect to addition. Thus if for some u ∈ M there exists v ∈ M such that u + v = 0 we have that u = u + 0 = u + (u + v) = (u + u) + v = u(1 + 1) + v = u + v = 0, proving our claim.
In view of Remark 2.2.29, any semialgebra (in particular a semifield) over a bipotent semifield is inverse free with respect to addition. i,j with a i,j ∈ A and k i,j ∈ N. D is said to be affine over H, or an affine extension of H, if A is finite. Note 2.2.32. Later in this section we introduce the notion of a 'semifield with a generator'. In this definition, we define generators of a semifield to be elements generating it as a kernel. In order to avoid ambiguity, when we refer to generation as in Definition 2.2.31 we will indicate it explicitly. In any other case we consider generation as a kernel. Definition 2.2.33. For any set X and any semiring S, F un(X, S) denotes the set of functions f : X → S. F un(X, S) also is a semiring, whose operations are given pointwise: for all a ∈ X. The unit element of F un(X, S) is the constant function always taking on the value 1 S . Remark 2.2.34. If S is a semifield without a zero element then F un(X, S) is a semifield (without a zero element). Indeed, for any f ∈ F un(X, S) we have that f −1 (x) = f (x) −1 is the inverse function of f .
Recall that we always assume a semifield does not contain a zero element, unless stated otherwise. are considered as functions defined over D n . The second way is to consider the variables x 1 , ..., x n as symbols rather than functions and taking the formal addition and multiplication operations on S[x 1 , ..., x n ].
Note 2.2.36. We always consider the polynomial semiring S[x 1 , ..., x n ] (and its semifield of fractions) mapped to the semiring of functions.
Example 2.2.37. For a nontrivial bipotent semifield H, the semiring H[x 1 , ..., x n ] (considered as a subsemiring of F un(H n , H)) is idempotent but not bipotent (for example the constant function α for any α ∈ H and the function x are incomparable). Since H is bipotent, for any pair of polynomials f, g ∈ H[x 1 , ..., x n ] we have that (f (a) + g(a)) m = f (a) m + g(a) m at any given point a ∈ H n . Thus (f + g) m = f m + g m globally over H n , i.e., as elements of H[x 1 , ..., x n ]. So H[x 1 , ..., x n ] satisfies the Frobenius property though it is not bipotent. Note that the arguments introduced above apply more generally to the semiring of functions yielding that the Frobenius property holds there too.

The structure of a semifield
Like a normal subgroup in group theory and an ideal in commutative ring theory, the kernel encapsulates relations on semifields. In the following few paragraphs, we introduce the notion of a kernel along with some of its properties. Definition 2.2.38. A subset K of a semifield H is a semifield-kernel of H if K is a normal subgroup in H with the convexity property that for every x, y ∈ H such that x + y = 1, a, b ∈ K ⇒ xa + by ∈ K. (2.5) The set of all the kernels of a semifield H is denoted by Con(H).
Note 2.2.39. We note that the name 'kernel' and the notation Con are customary in previous study of semifields. From now on we refer to a semifield-kernel simply as a 'kernel'. In places where confusion arises with the notion of a kernel of a homomorphism, we provide clarification.
• Some may find the name 'kernel' not necessarily the best choice for a name for the above structure. We presume the motivation for the name is that a 'kernel' is a kernel of a semifield homomorphism. It obviously might cause a little confusion. When such confusion may occur we explicitly indicate to which notion of kernel we refer.
• Though the so-called 'convexity' condition (2.5) may give a somewhat misleading impression, kernels are nothing but a special kind of groups inside the semifield H.
Remark 2.2.41. [9,Proposition 1.1] An equivalent definition of a kernel of a semifield H is the class [1] ρ of an arbitrary congruence ρ on H.
Remark 2.2.42. If S is an idempotent semifield since 1 + 1 = 1, we get that for any kernel K of S, a, b ∈ K ⇒ a + b = 1a + 1b ∈ K, yielding that K is itself a semifield. A particular case of interest is the semifield of fractions H(x 1 , ..., x n ) of H[x 1 , ..., x n ]. If H is idempotent, then H(x 1 , ..., x n ) is idempotent which yields that the kernels of H(x 1 , ..., x n ) are subsemifields of H(x 1 , ..., x n ).
Note 2.2.43. Throughout this dissertation, we work with an underlying semifield H of H(x 1 , ..., x n ) which is both idempotent and archimedean. In particular, under the assumption of idempotency, all kernels are also semifields. Theorem 2.2.44. [3,Theorem 3.6] The set Con(H) of all kernels of a semifield H forms a full modular lattice with respect to the operations of multiplication and the intersection of kernels, canonically isomorphic to the lattice of all possible congruencies on H.
Remark 2.2.45. [3] Let K 1 and K 2 be kernels of the semifield H. Then K 1 ∩K 2 and K 1 · K 2 are kernels of H. Moreover K 1 · K 2 is the smallest kernel in H containing Note 2.2.46. Note that multiplication of kernels is formulated by as customary in the theory of groups. Lemma 2.2.47. [9,Lemma 4.1] The following equalities hold for arbitrary kernels A, B and K of a semifield H, among which at least one is a semifield: Corollary 2.2.48. Since every kernel of an idempotent semifield is also a semifield, we have by Lemma 2.2.47 that its lattice of kernels is distributive. Thus every idempotent semifield is distributive.
The following are the three fundamental isomorphism theorems. Theorem 2.2.49. [3,Theorems 3.4 and 3.5] Let H 1 , H 2 be semifields and let R ⊂ H 1 be a subsemifield of H 1 . Let φ : H 1 → H 2 be a semiring homomorphism and let K be the homomorphism kernel of φ. Then the following hold: 1. φ(R) ⊂ H 2 is a subsemifield of H 2 . The homomorphism kernel of the re- 2. φ −1 (φ(R)) = KR which is a subsemifield of H 1 .
3. For any kernel L of H 1 , φ(L) is a kernel of φ(S 1 ).
4. For a kernel K of φ(H 1 ), φ −1 (K) is a kernel of H 1 . In particular, for any kernel L of H 1 we have that φ −1 (φ(L)) = K L is a kernel of H 1 .
Theorem 2.2.51. [11] Let H be a semifield and K a kernel of H.
1. If U is a subsemifield of H, then U ∩ K is a kernel of U and K a kernel of the subsemifield U · K = {u · k : u ∈ U, k ∈ K} of H and one has the isomorphism U/(U ∩ K) ∼ = U · K/K.

If
L is a kernel of H, then L ∩ K is a kernel of L and K a kernel of L · K.
Now, one has in general only the group isomorphism L/(L ∩ K) ∼ = L · K/K which is a semifield isomorphism exactly in the case when L is also a subsemifield of H.
Theorem 2.2.52. [11] Let H be a semifield and let K and L be kernels of H satisfying K ⊆ L. Then L/K is a kernel of H/K and one has the semifield isomorphism H/L ∼ = (H/K)/(L/K).
The following result concerns the induced order of the quotient semifield. It holds for any idempotent semifield S, and in particular for H(x 1 , ..., x n ) with H an idempotent semifield. Theorem 2.2.53. Let H be a semifield and let L ∈ Con(H) be a kernel of H. Every kernel of H/L has the form K/L for some kernel K ∈ Con(H) uniquely determined such that K ⊇ L, and there is a 1 : 1 correspondence {Kernels of H/L} → {Kernels of H containing L} given by K/L → K.
Proof. From the theory of groups we have that there is such a bijection for normal subgroups. To apply the theorem for kernels, we only need to show that a homomorphic image and preimage of a kernel are kernels, which in turn is true by Theorem 2.2.49. Remark 2.2.54. Let K be a kernel of an idempotent semifield S. Then the induced order on the quotient semifield S/K is such that aK ≤ bK ⇔ (1 + ab −1 ) ∈ K.
Proof. Let a, b be elements of H. The induced order on H/K is given by: aK ≤ bK if and only if there exists some c ∈ K such that a ≤ cb. Now, the following identity holds in H: a = (a ∧ b)(ab −1 ∔ 1) ( distributing the right hand side and the left hand side one at a time give opposite weak inequalities). Consequently, if (ab −1 ∔ 1) ∈ K then a = (a ∧ b)(ab −1 ∔ 1), implying that aK = (a ∧ b)K ≤ bK. Conversely, let c ∈ K such that a ≤ cb. Then we have (a We conclude this part with the definition of a large kernel.  Definition 2.3.1. Let A be a subset of a semifield H. Denote by A the smallest kernel in H containing A. It is equal to the intersection of all kernels in H containing A. If H = A , then A is called a set of generators of the semifield H (as a kernel).

Semifield with a generator and generation of kernels
A kernel K is said to be finitely generated if K = A where A is a finite set of elements of H. By Remark 2.2.45, if K is generated by {a 1 , ..., a t } ⊂ H then K = a 1 · · · · · a t (the smallest kernel containing {a 1 , ..., a t }). In such a case, we write K = a 1 , ..., a t to indicate that K is generated by {a 1 , ..., a t }. If K = a for some a ∈ H, then K is called a principal kernel. A semifield is said to be finitely generated if it is finitely generated as a kernel. If H = a for some a ∈ H, then a is called a generator of H and H is said to be a semifield with a generator. In other words, a semifield with a generator is a semifield which is principal as a kernel of itself.
Proof. Let a + a −1 ∈ K. Then in the factor semifield H/K, we get w + w −1 = 1 for each element w = aK which yields that w ≤ 1, w −1 ≤ 1, and thus w = 1, i.e., aK = K and so a ∈ K. For the second condition just apply Lemma 2.2.13. Proposition 2.3.3. [9] Let H be a semifield. Then for any a ∈ H such that the kernel generated by a is a semifield, the following equality holds: In words, the kernel generated by a coincides with the kernel generated by a+ a −1 .
Proof. A direct consequence of Lemma 2.3.2, which implies that a ∈ a + a −1 , and thus a ⊆ a + a −1 . The converse inclusion follows from the fact that a is a semifield.
a + u + v = c. The equality K + uK + vK = K holds in the factor semifield H/K = {yK : y ∈ H}. Then bK = (a + u)K = aK + uK = K by Lemma 2.2.13. Thus b ∈ K.
Proposition 2.3.5. Let N be a convex normal subgroup of a semifield H. If each a ∈ N is comparable with 1, i.e., if ≤ is a total order on N, then N is a kernel of H.
Proof. In order to prove N is a kernel, we need to show that for a ∈ N and s, t ∈ H such that s + t = 1, s + th ∈ N. By assumption, we have that 1 ≤ a or a ≤ 1. In the former case, we have t ≤ ta and s ≤ sa, thus 1 = s + t ≤ s + ta ≤ sa + ta = (s + t)a = a. Since N is convex and a, 1 ∈ N we get that s + ta ∈ N. The latter case yields that a ≤ s + ta ≤ 1 which implies the same.
The following remark yields an important property of a kernel, which we call 'power-radicality', to be introduced shortly.
Remark 2.3.6. The multiplicative group of every proper semifield H is a torsionfree group, i.e., all of its elements that are not equal to 1 have infinite order.
The following remark is a straightforward consequence of Remark 2.3.6. Remark 2.3.7. Let K be a kernel of a proper semifield H. For every a ∈ H, if a n ∈ K for some n ∈ N then a ∈ K. We refer to this property of kernels by saying that a kernel is power-radical.
Proof. Indeed, if there exists an element a ∈ H not admitting the stated property, then its image φ(a) in H/K under the quotient homomorphism is torsion, which by Remark 2.3.6 is not possible since H/K is a semifield.
The following subsequent statements establish a connection between the (normal) group generated by a set of elements and the kernel generated by the set. Recall that, kernels are a specific kind of group, the (normal) subgroup generated by a set of elements need not be a kernel. Proposition 2.3.8. [3,Proposition (3.13)] Let H be a semifield and let N be a normal subgroup of (H, ·). Then the smallest kernel containing N is given by Proof. As a kernel is defined to be a multiplicative (normal) group, the assertion is immediate. Remark 2.3.10. Let S 1 , ..., S r ⊂ H and let G 1 , ..., G r be the groups generated by S 1 , ..., S r respectively. Then is a kernel, and thus a group, which contains all the groups G i for i = 1, ..., r, thus also contains the group r Remark 2.2.45), we get the converse inclusion and thus equality. All other equalities are group theoretic basic equalities. Proposition 2.3.11. [9,Proposition (3.1)] Let K = a , a principal kernel in a semifield H with a ∈ H such that a ≥ 1. Then K = {x ∈ H : ∃n ∈ N such that a −n ≤ x ≤ a n }.
Proof. This is a direct consequence of Proposition 2.3.11 and Proposition 2.3.3.
Note 2.3.15. Note that for an idempotent semifield the equality introduced in Corollary 2.3.14 can be restated as (2.14) Proof. Just take inverses in equation (a + a −1 ) −n ≤ x ≤ (a + a −1 ) n and sum up both sides of resulting weak inequalities.  [9,Property (3.2)] The homomorphic image of a generator is a generator of the image. In particular, a homomorphic image of a semifield with a generator is a semifield with a generator.
Proof. If φ : H → U is a homomorphism of a semifield H onto a semifield U and H = a , then U = φ(a) , because the preimage of a kernel at a homomorphism of semifields is always a kernel. Remark 2.3.19. Let a be a principal kernel of a semifield H, which is also a semifield and let φ : H → U be a semifield epimorphism. Then i.e., the image of a is the kernel generated by φ(a) in U.
Proof. As a is also a semifield, it is a semifield with a generator a. Thus by Remark 2.3.18 its homomorphic image is also a semifield with a generator φ(a) and by Theorem 2.2.49 it is a kernel. Thus, its homomorphic image is a principal kernel φ(a) which is also a semifield.

Proof. Let H(x) be the semifield of fractions of H[x]
where H is a bipotent semifield. We will now show that x is a maximal kernel of H(x). Consider the and thus ψ(a) = 1 if and only if m i=1 α i = k j=1 β j . We will show that if ψ(a) = 1 then a ∈ x . Indeed, assume ψ(a) = 1, denote α = m i=1 α i , β = k j=1 β j , then by the above α = β. We have By assumption α β = 1, and thus a ∈ x as desired.
As H is a bipotent semifield and thus completely ordered and hence simple, we can use the assertions above and Corollary 2.5.8, to deduce that x is a maximal kernel of H(x).
Applying a change of variable y = x γ for each γ ∈ H, the above proof implies that the kernel x γ is the kernel of the substitution map ψ γ : H(x) → H defined by x → γ. Since H = {1} is simple it is generated by any γ ∈ H so ψ γ is onto and consequently, by the same argument used above, x γ is maximal for any γ ∈ H.
Intersecting both sides of the last equality with H , we get for any α ∈ R \ {1}.

Distributive semifields
As will be shown in the section concerning idempotent semifields and latticeordered groups, the lattice of kernels every of an idempotent semifield is distributive. Idempotent semifields play an important role in our theory. Here we introduce some of the properties of semifields which have a distributive lattice of kernels.
2. All proper kernels of H are intersections of its irreducible kernels. As every reducible kernel is weakly irreducible, the first assertion of Proposition 2.6.2 states that Note 2.6.3. In a distributive semifield H, a kernel K of H is irreducible if and only if for any pair of kernels A, B of H.
The following is a nice example of utilizing of irreducible kernels.
Remark 2.6.4. Let H be a distributive semifield. Then, for any a, b ∈ H such that Proof. As ab ∈ a b , obviously ab ⊆ a b . Since a ∩ b = {1}, we have that a ∈ P or b ∈ P for any P ∈ Spec(H) (by definition of irreducibility of a kernel and the fact that {1} ⊆ P ). If ab ∈ P , then if a ∈ P we get b = a −1 (ab) ∈ P and if b ∈ P we get a = (ab)b −1 ∈ P . In any case both a and b are in P and thus we have shown that ab ⊆ P implies a b ⊆ P . Now, as H is distributive, we have by Proposition 2.6.2 (1) that a b = ab as they are both intersections of the irreducible kernels containing them.
Kernels having trivial intersection ( {1} ) play a very important role in our theory.
Remark 2.6.5. [9] Let H be a distributive semifield. Then any proper kernel of H is the intersection of all irreducible kernels containing it. In particular, the intersection of all the irreducible kernels of H equals {1}.

Idempotent semifields : Part 1
In the following subsection, we concentrate our attention on the theory of idempotent semifields. We will continue the study of idempotent in subsequent sections after introducing the theory of lattice-ordered groups.
It turns out that this special kind of semifield has some very nice additional properties to those of general semifields that make it quite easy to work with.
As we have already noted, kernels of an idempotent semifield are themselves semifields. We now introduce a few more interesting properties of such semifields.
Remark 2.7.1. Finitely generated kernels of an idempotent semifield are principal.
Proof. This follows directly from Theorem 2.3.22, as a kernel of an idempotent semifield is itself a semifield.
Remark 2.7.2. Since every finitely generated kernel of an idempotent semifield is itself a semifield with a generator, by Corollary 2.3.12, we have that for any such kernel we can choose a generator a such that |a| = a ≥ 1. This issue is important, as for any such an element 1 ∔ a = sup(1, a) = max(1, a) = a and 1 ∧ a = inf(1, a) = min(1, a) = 1, which imply that 1 ∔ (a −1 ) = sup(1, a −1 ) = max(1, a −1 ) = 1 and that 1 ∧ (a −1 ) = inf(1, a −1 ) = min(1, a −1 ) = a −1 . By convention, we always take a generator of a kernel of an idempotent semifield to be such a 'positive' generator unless stated otherwise. The importance of such generator is that it is comparable to 1.
Remark 2.7.3. For any kernel K of an idempotent semifield the following holds: Proof. Indeed, Since |g|, |h| ≥ 1, we have that |g| ≤ |g||h| and |h| ≤ |g||h| thus On the other hand, we have that Proof. First note that as e · n i=1 x i is closed under multiplication and addition (since it is a semifield), by Remark 2.3.16 it is enough to prove that for any monomial f in H[x 1 , ..., x n ] there exists some k ∈ N such that Since H is a semifield with a generator, and e is a generator of H, we have that α ∈ e . Thus there exists some s such that |α| ≤ |e| s . Now, as |x p 1 1 · · · · · x pn n | ≤ |x 1 ∔ · · · ∔ x n | |p 1 |∔···∔|pn| = |x 1 | |p 1 |+···+|pn| ∔ · · · ∔ |x n | |p 1 |+···+|pn| . For k = max(s, |p 1 | + · · · + |p n |), we get where I ⊂ Z containing n elements. First, note that since H is idempotent (1 + 1 = 1), x is closed with respect to addition and thus a semifield. Now, n i=1 f i (x) = 1 yields by the natural order that f i ≤ 1 for every i = 1, ..., n.
For simplicity of notation, denote x 0 = e, where as above e is the generator of H as a kernel over itself.
The exact same construction yields that the kernel (2.25) where I ⊂ Z containing n elements. Notice that in the special case where e = 1 the kernel in (2.25) degenerates to {1}. Further, (2.25) applies to any element f (x 1 , ..., x n ) of H(x 1 , ..., x n ) taken in place of x k . Now, for each x i , i = 0, 1, ...., n since x i are principal kernels and since none of the elements in {x 0 = e, x 1 , ..., x n } are comparable to each other, Proposition 2.3.11 yields that x i ∈ x j for any j = i. Thus, none of these kernels is contained in another.
The collection PSpec(R(x 1 , ..., x n )) of all irreducible principal kernels of the semi- It is a topological space with respect to the principal Stone topology. Its subspace PMax(R(x 1 , ..., x n )), consisting of all maximal principal kernels, is called the maximal principal spectrum.
Remark 2.7.17. The Stone topology on Con(R(x 1 , ..., x n )) gives rise to the induced topology on the kernels of the semifield with a generator R , the kernels of which, Con( R ), form a lattice of kernels which embeds as a sublattice of Con(R(x 1 , ..., x n )).
Remark 2.7.18. In the next section, we show that which forms a sublattice of PCon(R(x 1 , ..., x n ). Thus the principal Stone topology induces a topology on PCon( R ).

Affine extensions of idempotent archimedean semifields
In the following, we characterize the affine idempotent semifield extensions of an idempotent archimedean semifield H which are divisible over H (divisible extensions are the analogue for algebraic extensions in ring theory). Proof. For a, b ∈ H there exist some (minimal) k, m ∈ N such that a k = α, b m = β are elements of H. First note that for every α, β ∈ H and t ∈ N, by the former observation. Since both α m ∈ H and β k ∈ H we have that the Frobenius property holds, so that (α m + β k ) t = α mt + β kt = a kmt + b kmt . As a kmt + b kmt ≤ (a t + b t ) km , we conclude that (a s b t−s ) km ≤ (a t + b t ) km which implies that a s b t−s ≤ a t + b t for any 0 ≤ s ≤ t (since H is archimedean) and so we get that (a + b) t = t s=0 a s b t−s = a t + b t , as desired.
Remark 2.8.4. Let H be an idempotent semifield satisfying the Frobenius property. Let K = H(a 1 , ..., a n ) be an affine semifield extension of H with a 1 , ..., a n ∈ H. The substitution map φ : K is a principal kernel. We claim that Kerφ = K.
An affine semifield over an idempotent semifield H, in particular, the semifield of fractions H(x 1 , ..., x n ), is an idempotent semifield. It turns out that H(x 1 , ..., x n ) and generally every idempotent semifield can be considered as a special kind of group, called a lattice-ordered group, or ℓ-group, in which the notion of kernels coincides with the notion of normal and convex ℓ-subgroups.
Although we consider H(x 1 , ..., x n ), all results given below hold for any idempotent semifield, in particular for the kernels of H(x 1 , ..., x n ) each of which is an idempotent semifield in its own right. A particularly important such kernel is H which denotes the kernel of H(x 1 , ..., x n ) generated by any α ∈ H \ {1}.
Due to the central role idempotent semifields play in our theory, we hereby introduce some basic notions and some important results in the theory of latticeordered groups. All relevant definitions can be found in [8]. We note that in [8] the group operation is taken to be addition while in our context we take it to be multiplication, as we use ∔ for sup , i.e., ∨. The order preserving map x → a x with a a symbol is used to translate the statements presented there to our language where the group is taken to be multiplicative. In particular (R, +) is translated to (R + , ·).

Lattice-ordered groups
Definition 3.1.1. A partially-ordered group (p.o group) is a group G endowed with a partial order such that the group operation preserves the order on G, that is, a ≤ b ⇒ ∀g ∈ G (ag ≤ bg and ga ≤ gb).
If the partial order on G is directed, then G is a directed group. If the partial order on G is a lattice order, then G is a lattice-ordered group or ℓ-group. If the order on G is a linear order, then G is called a totally-ordered group or o-group.
2. An ℓ-subgroup of a po-group is a subgroup which is also a sublattice.
Definition 3.1.4. If G and H are ℓ-groups, then a group homomorphism φ : G → H which is also a lattice homomorphism (preserves ∧ and ∨ (∔ in our notation)) is called an ℓ-homomorphism.
Definition 3.1.5. A subset S of a poset P is said to be convex if a ≤ p ≤ b with a, b ∈ S implies that p ∈ S.
Remark 3.1.6. [8,Theorem (2.2.3)] All fundamental group isomorphism theorems hold for normal convex ℓ-subgroups of an ℓ-group.    3.1.9. [8,Theorem (2.2.5)] Let G be an ℓ-group. The lattice of convex ℓ-subgroups of G is a complete distributive sublattice of the lattice of subgroups of G, and, in fact, it satisfies the infinite distributive law Definition 3.1.10. An ℓ-group G is a subdirect product of the family {G i : i ∈ I} of ℓ-groups if there is a monomorphism φ : G → G i such that each composite π i • φ is an epimorphism, and G is subdirectly irreducible if, in any such representation of G there is an index i such that π i • φ is an isomorphism. We indicate that G is a subdirect product of the family {G i : i ∈ I} by writing Note 3.1.11. We use the same notation presented in Definition 3.1.10 in the context of idempotent semifields.
Remark 3.1.12. A function φ : G → G i on an ℓ-group G is uniquely determined by the family {π i • φ : i ∈ I}, and is an ℓ-homomorphism if and only if each π i • φ is an ℓ-homomorphism. In this case Kerφ = i Ker(π i • φ) and G/Ker(π i • φ) ∼ = Im (π i • φ). Consequently, each family {N i : i ∈ I} of normal convex ℓ-subgroups of the ℓ-group G determines a homomorphism G → G/N i with kernel N = i N i , and G/N is a subdirect product of the family {N i : i ∈ I} and all subdirect product representations of G/N essentially arise in this way. Clearly, a nonzero ℓ-group G is subdirectly irreducible if and only if it has a smallest nontrivial (i.e., = {1}) normal convex ℓ-subgroup.
Proof. Let G be an ℓ-group. For a ∈ G \ {1} let N a be a normal convex ℓ-group of G which is maximal with respect to excluding a. The existence of N a is given by Zorns Lemma. Since each normal convex ℓ-subgroup of G that properly contains N a must contain a, G/N a is subdirectly irreducible (it has a smallest nontrivial (contains the coset aN a ) normal convex ℓ-subgroup. But a =1 N a = {1}, so G is isomorphic to a subdirect product of the G/N a . Shortly we will introduce a much stronger result for commutative ℓ-groups. Definition 3.1.14. [8,Section (2.4)] A convex ℓ-subgroup P of the ℓ-group G is called a prime subgroup if whenever a, b ∈ G with a ∧ b ∈ P , then a ∈ P or b ∈ P .
Note 3.1.15. In the context of idempotent semifields, we use the notion 'irreducible' for 'prime'. Proposition 3.1.16. [8,Theorem (2.4.1)] The following statements are equivalent for the convex ℓ-subgroup P of G: 3. The lattice of (left) cosets G/P is totally ordered.
4. The lattice of convex ℓ-subgroups of G that contain P is totally ordered.
Remark 3.1.17. [8] Each subgroup L that contains a prime subgroup is an ℓsubgroup, and hence is itself prime if it is convex. For if a ∈ L, then, since (a ∨ 1) ∧ (a −1 ∨ 1) = 1, (a ∨ 1) = a(a −1 ∨ 1) ∈ L and so by Remark 3.1.3 L is an ℓ-subgroup. It can be easily seen that the intersection of any chain of prime subgroups is prime. In particular, if P is a prime subgroup and {P } is enlarged to a maximal chain {P i : i ∈ I} of prime subgroups, then P is a minimal prime contained in P . Proposition 3.1.18. Each convex ℓ-subgroup is the intersection of prime subgroups.
Definition 3.1.19. Let G be an ℓ-group. For a, b ∈ G, a and b are said to be disjoint or orthogonal if |a| ∧ |b| = 1.
Remark 3.1.20. Let G be a commutative ℓ-group. For any a ∈ G, the set is a convex ℓ-subgroup of G.
Theorem 3.1.21. A commutative ℓ-group is either linearly ordered or subdirectly reducible.

Idempotent semifields versus lattice-ordered groups
Endowed with the natural order given in Remark 2.2.14, an idempotent semifield of fractions H(x 1 , ..., x n ) can be considered as a ℓ-group. The following results establish the connection between ℓ-groups and additively commutative and idempotent semifields in general, and in particular H(x 1 , ..., x n ).
There exists a correspondence between ℓ-groups and additively commutative and idempotent semifields.
The following is a direct consequence of Remark 3.1.8: Let H be an idempotent commutative semifield. If K is a kernel of H and L is a kernel of K (viewing K as an idempotent semifield) then L is a kernel of H.
In view of the above, all statements presented in 3.2.1 and 3.2.2 referring to ℓ-groups are true for idempotent semifields, changing notation from ∨ to ∔.

Idempotent semifields: Part 2
The structure of lattice-ordered groups is thoroughly studied in [6] and [1], and many results given there are applicable for a semifield of fractions H(x 1 , ..., x n ) with H a bipotent semifield and generally for affine semifield extensions of H (which are just quotients of H(x 1 , ..., x n ) by Remark 2.3.23).
The results we introduce in the remainder of this section are derived from known results in the theory of lattice-ordered groups.
The notions of an ℓ-group and normal convex ℓ-subgroups correspond to an idempotent semifield and their kernels. In our case the idempotent semifield is also commutative, thus normality of an ℓ-subgroup is insignificant. In the special case of the idempotent semifield H(x 1 , ..., x n ), we call H(x 1 , ..., x n ) + and H(x 1 , ..., x n ) − the positive cone of fractions and the negative cone of fractions and denote them by P + and P − , respectively.
As the following Theorem demonstrates, the partial orders of the group (H(x 1 , ..., x n ), ·) that make it into a po-group are in one-to-one correspondence with the positive cones of H(x 1 , ..., x n ).
Conversely, if P is a (normal) sub semigroup of S that satisfies (2), then the relation defined in (3) is a partial order of S which makes S into a partially ordered group with positive cone P .  • G is totally ordered if and only if G = G + ∪ G − .
• G is directed if and only if G + generates G. Moreover, if G + generates G, • G is an ℓ-group if and only if G + is a lattice and generates G as a group. Corollary 3.3.5. P + is a lattice and generates H(x 1 , ..., x n ).
Proof. Note that P + being a lattice means that for any f, g ∈ P + both f ∧ g and f ∔ g (i.e., f ∨ g) are in P + . The assertion follows directly from Theorem 3.3.4. Definition 3.3.6. A lattice L is infinitely distributive if whenever {x i } is a subset of L for which x i exists, then, for each y ∈ L, (y ∧ x i ) exists and The dual also holds.
, then both sides of the equality are equal to f (x). If g(x) ≥ h(x) ≥ f (x) then both sides of the equality are equal to h(x). If h(x) ≥ g(x) ≥ f (x) then both sides of the equality are equal to g(x).
As the above holds for every x ∈ H n , (3.1) is an identity.
The first equality from the right follows the assumption that f ≥ |g| ∧ |h|, the second follows Lemma 3.3.9 and the last follows the assumptions |f | ≤ |g| and |f | ≤ |h|. and By definition |f | ≥ , |f | ≤ ∈ H(x 1 , ..., x n ).
Proof. We will only give the proof for the Riesz decomposition property. For all other properties see [8] (2.1.4).
Remark 3.3.16. For any f, g ∈ H(x 1 , ..., x n ) the following relations hold: Proof. See [6], Chapter 2. We note that the third statement is true due to the commutativity of H(x 1 , ..., x n ).

Structure of affine extensions of a bipotent semifield
Due to the correspondence between idempotent semifields and ℓ-groups, we can apply Theorems 3.1.21 and 3.1.22 to H(x 1 , ..., x n ) and deduce Proof. First note that any kernel of an idempotent semifield is itself an idempotent semifield. Now, linearly (totally) ordered ℓ-groups have no proper convex ℓ-subgroups and thus correspond to semifield having no proper kernels, i.e., simple semifields (see Remark 2.4.2). Remark 3.3.18. [5,Lemmas (3.1.8), (3.1.16)] [8,Sec. 2.1] For a torsion free partially ordered abelian group G there exists a group G which is the smallest divisible group containing G extending its order. If G is lattice ordered, directed or totally ordered then so is G, lattice ordered, directed or totally ordered, respectively. G is called the divisible hull of G. Definition 3.3.19. Viewing an idempotent semifield H as an ℓ-group, we define the divisible closure H of H to be its divisible hull. By the above, H is also an idempotent semifield (as it is an ℓ-group).
The following two results will be considered and proved in our subsequent discussions: Proposition 3.3.20. Let H be a bipotent semifield. As we have previously shown, any bipotent semifield is totally (linearly) ordered and thus a simple semifield. For each point a = (α 1 , ..., α n ) ∈ H n , let and Each Q a is an affine bipotent (thus totally ordered) semifield extension of H namely H(α 1 , ..., α n ) which is a simple. In the case where H is divisible, we have each Q a ∼ = H.
Proof. Let f ∈ L = a∈H n L a . Then for each a ∈ H n since f ∈ L a we have that

Archimedean idempotent semifields
All the statements introduced in this section were stated originally for additive ℓ-groups and have been translated by us to the language of commutative idempotents semifields, where the group operation is multiplication instead of addition.
Recall that an idempotent semifield H is said to be archimedean if it is archimedean as a po-group, i.e., for a, b ∈ H if a Z ≤ b then a = 1.
Proof. Let f, g ∈ H(x 1 , ..., x n ) such that f Z ≤ g . If a ∈ H n then by our assumption f (a) k ≤ g(a) for all k ∈ Z. Since H is archimedean and f (a), g(a) ∈ H, we have that f (a) = 1. Since this holds for any a ∈ H n we have f (H n ) = 1 implying f = 1. The arguments above hold for any kernel of H(x 1 , ..., x n ). Proof. Let X ⊂ F un(H n , H) be bounded from below, say by h ∈ F un(H n , H). Then for any a ∈ X the set {f (a) : f ∈ X} is bounded from below by h(a) thus has an infimum f ∈X f (a). It is readily seen that the function g ∈ F un(H n , H) defined by g(a) = f ∈X f (a) is an infimum for X, i.e., g = f ∈X f . Analogously, if X is bounded from above then ( f ∈X f )(a) = f ∈X f (a) is the supremum of X.
Definition 3.4.5. A completion of the idempotent semifield H is a pair (H, θ) where H is a complete idempotent semifield and θ : H → H is a monomorphism whose image is dense in H.
The following theorem states that each archimedean idempotent semifield has a unique completion. Theorem 3.4.6. [8,Theorem 2.3.4] An idempotent semifield has a completion if and only if it is archimedean. If (A, θ A ) and (B, θ B ) are two completions of the idempotent semifield H, then there is a unique isomorphism ρ : Definition 3.4.7. A subset A of the poset P is called co-final in P if for every x ∈ P there exists some a ∈ A such that x ≤ a. A is said to be co-initial in P if for every x ∈ P there exists some a ∈ A such that a ≤ x.
Theorem 3.4.9. [8,Theorem (2.3.6)] Let H be an archimedean subsemifield of the complete idempotent semifield H. Then the following statements are equivalent: 1. H is the completion of H. We now state the well-known result by Hölder for ℓ-groups: Theorem 3.4.11. [8,Theorem (2.3.10)(Hölder)] The following statements are equivalent for an ℓ-group G.

H is left dense in H, and if
1. The only convex ℓ-subgroups of G are 1 and G.
2. G is totally ordered and archimedean.
Proposition 3.4.12. [8] A divisible totally ordered archimedean group which is complete is isomorphic to (R, +).
Translating the above to the language of idempotent semifields and using the isomorphism (R, +) → (R + , ·) defined by x → e x (where R + is the set of positive real numbers) yield Theorem 3.4.13. The following statements are equivalent for an idempotent semifield H.
In view of the above, we may regard the designated semifield R as being (R + , ∔, ·).

Skeletons and kernels of skeletons
In this section we introduce the notions of 'skeletons' and 'kernels of skeletons'.
We define a pair of operators Skel and Ker where Skel maps a kernel to its skeleton and Ker maps a skeleton to its corresponding kernel. Although we define these operators with respect to the semifield of fractions H(x 1 , ..., x n ), since we only use the fact that H(x 1 , ..., x n ) is a semifield to prove our assertions, all the properties proved in subsections 4.1 and 4.2 hold replacing H(x 1 , ..., x n ) by any subsemifield of the semifield of functions F un(H n , H).

Skeletons
We now define a geometric object in the semifield H n to which we aim to associate to a kernel of the semifield of fractions H(x 1 , ..., x n ). Namely, we define an operator Skel : P(H(x 1 , ..., x n )) → H n which associates a subset of H n to every kernel of H(x 1 , ..., x n ).
Note 4.1.1. In the field of computer vision the notion of skeleton (or topological skeleton) of a shape is a thin version of that shape that is equidistant to its boundaries.
Resembling the tropical variety in its shape, we have decided to call the next geometric object , which will be shown to generalize the notion of tropical variety, a 'skeleton'.
.., a n ) ∈ H n : f (a 1 , ..., a n ) = 1, ∀f ∈ S}. (4.1) ., x n ) the following statements hold: 3. i∈I Skel(S i ) = Skel( i∈I S i ) for any index set I and in particular, Proof. The first statement is set theoretically obvious and since S 1 ⊆ S 1 , it implies that Skel( S 1 ) ⊆ Skel(S 1 ) in the second assertion. To prove the opposite = 1 thus proving the skeleton of the semifield generated by S 1 is in The converse direction is obvious.
Remark 4.1.6. For H a bipotent semifield, the following statements hold for f, g ∈ H(x 1 , ..., x n ): have that sup(f (x), g(x)) = 1 if and only if f (x) = 1 and g(x) = 1. Analogously the same holds for f (x)g(x) = 1 when both f, g ≥ 1. ( Definition 4.1.7. Denote the collection of skeletons in H n by Skl(H n ) and the collection of principal skeletons in H n by P Skl(H n ).

Kernels of skeletons
In the following discussion, we construct an operator Ker : H n → P(H(x 1 , ..., x n )) which associates a kernel of the semifield of fractions H(x 1 , ..., x n ) to any skeleton in Skl(H n ). Then we will proceed to study the relation between this operator Ker and the operator Skel defined in the previous subsection. Throughout this section, we take H to be a bipotent (totally ordered) semifield. Ker(Z) = {f ∈ H(x 1 , ..., x n ) : f (a 1 , ..., a n ) = 1, ∀(a 1 , ..., a n ) ∈ Z} (4.2) Remark 4.2.2. For Z, Z i ⊂ H n the following statements hold: Proof. The first assertion follows from the proof of (2) of Proposition 4.1.5. The second assertion is a trivial set theoretic fact, which in turn implies that Ker( i∈I Z i ) ⊆ i∈I Ker(Z i ). The second inclusion of (3) is trivial. Assertions (4) and (5) are straightforward consequences of the definitions of Skel and Ker.
where Z is a suitable skeleton.
By the above assertions we have given by Z → Ker(Z); the reverse map is given by K → Skel(K).
Proposition 4.2.6. Let E 1 and E 2 be kernels in H(x 1 , ..., x n ), and let S 1 = Skel(E 1 ) and S 2 = Skel(E 2 ) be their corresponding skeletons. Then the following statements hold: Proof. For the first assertion, denote K = E 1 · E 2 . Since E 1 and E 2 are kernels, by Remark 2.2.45 K also is a kernel and E 1 , and so, by Proposition 4.1.5 (2), In view of the above we have that Now, Let x ∈ Skel(K) and assume to the contrary that x ∈ Skel(E 1 ) and As a special case of Proposition 4.2.6 we have Note that by convention Skel(f ) = Skel( f ).
5 The structure of the semifield of fractions 5.1 Bounded rational functions in the semifield of fractions Throughout this section we assume H to be a bipotent and divisible semifield. Any supplemental assumptions regarding H will be explicitly stated.
We begin by introducing an example which motivates our subsequent discussion in this section.
Example 5.1.1. Consider the principal kernel x . Its corresponding skeleton is defined by the equation Since min(|x|, |α|) = 1 ⇔ |x| = 1 we get that |x| ∧ |α| defines exactly the same skeleton as x . As x ∈ x ∩ α (x cannot be bounded by a bounded function) we have that x ⊃ x ∩ α . The cause of this phenomenon is the kernels of the form α with α ∈ R \ {1}.
These kernels are not kernels of R-homomorphisms, since for any such homomorphism φ, one must have α = αφ(1) = φ(α) but as α ∈ Kerφ we have that φ(α) = 1 too. Thus kernels containing them also fail to be kernels of Rhomomorphisms.
Proof. If f ⊇ α then in particular α ∈ f thus there exists some k ∈ N such that |α| ≤ |f | k thus β ≤ |f | for β ∈ H such that β k = |α| (such β exists as H is divisibly closed). Thus f is bounded from below. Conversely, if f is bounded from below, then there exists some α > 1 in H such that |f | ≥ α = |α| which yields that α ∈ f and thus α ⊆ f . Remark 5.1.4. Let f be a principal kernel such that f is bounded from below.
Then any generator g ∈ f is bounded from below.
Proof. Since g is a generator, we in particular have that f ∈ g . Then, By Remark 2.3.16, there exists some k ∈ N such that |f | ≤ |g| k . Since f is bounded from below there exists some α > 1 in H such that |f | ≥ α, thus |g| k ≥ |f | ≥ α, which yields that g is bounded from below by β ∈ H such that β k = α (and so β > 1).
Let H be a divisible bipotent semifield. We will begin by characterizing the Conversely, as the skeleton is determined by any generator of f , in particular |f |, Skel(f ) = ∅ implies that there is no solutions to the equation |f (x)| = 1. Since |f (x)| ≥ 1 for every x ∈ H n , this implies that |f (x)| > 1 for every x ∈ H n . Proposition 5.1.7. Let f be a principal kernel in H(x 1 , ..., x n ). Then the following statements are equivalent: 3. There exists a generator of f which is bounded from below. where h = h i 0 and g = g j 0 . Each region R is either closed and bounded or unbounded. In the former case |f |, being continuous, attains a minimum value over R; thus the image of |f | is bounded from below by some γ > 1. In the latter case, if l is constant then it trivially attains a minimal value and is bounded from below by some γ > 1. The only possibility we are left with is that l is not constant. Assume the image of l is not bounded from below by any γ > 1. Then there exists a one dimensional curve over which l is monotonically decreasing in a constant rate and thus must obtain the value 1 contradicting the assumption that |f (x)| > 1 for all x ∈ H n . ( It is convenient to consider l in logarithmic scale so that l takes the form of a linear form and the curve is a one dimensional affine space (a straight line) over which l has a constant slope.) Then Proof. If Skel( f ) = ∅ then by Proposition 5.1.7 f is generated by a bounded from below element, thus is a bounded from below kernel. Moreover, by Remark 5.1.4 we have that any generator of f is bounded from below. Conversely, if f is a bounded from below kernel, then f ′ is bounded from below for any generator f ′ of f , in particular for f ′ = f . Thus there exists some α > 1 in H such that |f | ≥ α, thus, by Remark 5.1.6, we have that Skel( f ) = ∅.
Definition 5.1.9. f ∈ H(x 1 , ..., x n ) is said to be bounded from above (or simply bounded ) if there exists some α ≥ 1 in H such that |f | ≤ α.
Remark 5.1.10. Let f be a principal kernel such that f is bounded from above. Then any g ∈ f is bounded from above. In particular, any generator of f is bounded from above.
Proof. By Remark 2.3.16 for any g ∈ f there exists some k ∈ N such that |g| ≤ |f | k . Since f is bounded there exists some α ≥ 1 in H such that |f | ≤ α, thus |g| ≤ |f | k ≤ α k , which yields that g is bounded.
Definition 5.1.11. A principal kernel f of H(x 1 , ..., x n ) is said to be bounded from above if it is generated by a function bounded from above. Proof. If f ⊆ α then in particular f ∈ α . Thus there exists some k ∈ N such that |f | ≤ |α| k , thus f is bounded from above. Conversely, if f is bounded from above, then there exists some α ∈ H ( α ≥ 1) such that |f | ≤ α = |α| which yields that f ∈ α and thus f ⊆ α . Proof. If f is bounded from below, then there exists some α > 1 in H such that |f | ≥ α thus by Remark 2.3.16 we have that α ∈ f . Since any α = 1 is a generator of H, we get that Proof. Let f ∈ H(x 1 , ..., x n ) be bounded from below. Then there exists some Let f be bounded from above. Then there exists some α ≥ 1 such that |f |∔α = α.
Then φ sends bounded from above principal kernels to bounded from above principal kernels and bounded from below principal kernels to bounded from below principal kernels.
Proof. The image of a kernel under a homomorphism is a kernel. Now, since φ( f ) = φ(f ) and since a principal kernel is bounded from above or bounded from below if and only if its generator is, the result follows from Remark 5.1.14.
Proof. First note that for α ∈ H such that α = 1, α generates H as a semifield and thus α = H . The assertion follows from Remark 5.1.12, since f ∈ α = H if and only if f ⊆ α .
As we shall now show, there is a close connection between bounded from above and bounded from below kernels: In view of the preceding discussion in this section, f · H is the smallest bounded from below kernel containing f while f ∩ H is the largest bounded from above kernel contained in f . One can consider the quotient f /( f ∩ H ) as a measure for "how far a kernel is from being bounded". The kernel R plays an important role in our theory. It actually contains all the kernels needed to form a correspondence between principal kernels and principal skeletons. In essence, every principal kernel f in PCon(R(x 1 , ..., x n )) has an bounded from above "copy" in the family of principal kernels contained in R , which possesses the exact same skeleton. Our so called "Zariski correspondence" will take place between the principal kernels contained in R and the skeletons in R n .

The structure of H
Let H be a bipotent divisible archimedean semifield. In the following discussion, we study the structure of the subsemifield and kernel H of H(x 1 , ..., x n ), which by Remark 2.3.25 is generated, as a kernel, by any α ∈ H \ {1}. As we have shown in Remark 5.1.16, H is comprised of bounded from above elements of H(x 1 , ..., x n ). Note that H is always assumed to be a bipotent divisible semifield while any additional assumptions regarding H will be stated when necessary.
For any principal kernel f ∈ PCon( H ) bounded from above, there exists an unbounded from above kernel In particular, Proof. Let f (x 1 , ..., x n ) ∈ H be bounded from above. Then there exists some As |f | is continuous we have that α i = α are all the same. Now define the following function Then for every a ∈ S, f ′ (a) = g(a) + |α|. Moreover, for every b = (b 1 , ..., b n ) ∈ S there exists some j such that |b j | ≤ β thus we have that (β −1 |b 1 |+....+β −1 |b n |) ≤ 1 Proof. By Proposition 5.2.1 for every f bounded from above there exists such f ′ .
On the other hand, if f is not bounded from above, take f ′ = f .
The following definition generalizes the notion of principal kernels bounded from above and from below introduced earlier.
By definition 5.2.3 every kernel K ⊆ H is a bounded kernel and vice versa.

Proof. For any
Thus (B, ·, ∩) is a sublattice of (Con(H(x 1 , ..., x n )), ·, ∩). Consequently, Proof. The first statement follows from the equality f ∩ H = |f | ∧ |α| and Remark 5.2.8. The following hold for the kernels of the semifield H : Moreover, every kernel of H is also a kernel of H(x 1 , ..., x n ).
Proof. Since H is a subsemifield of H(x 1 , ..., x n ) by Theorem 2.2.51(1) for any Conversely, as H(x 1 , ..., x n ) is idempotent and commutative, by Remark 3.2.3 we have that any kernel L of H is a kernel of H(x 1 , ..., x n ). Moreover, since L ⊆ H we have that L = L ∩ H . The second equality holds since for any principal kernels f ∈ PCon(H(x 1 , ..., x n )), f ∩ H = |f | ∧ |α| ∈ PCon( H ).

The polar-skeleton correspondence
In subsection 4.2 we introduced the notion of a K-kernels of H(x 1 , ..., x n ) whom have been shown to correspond to skeletons in H n (Proposition 4.2.5). Our next aim is to characterize these special kind of kernels. In analogy to algebraic geometry, we are looking for our 'radical ideals' which are shown to correspond to algebraic sets (zero sets) by the Nullstellensatz theorem. Unfortunately in our context 'radicality' does help much since by Remark 2.3.7 every kernel is powerradical. It actually turns out that in our theory, the role of radical ideals is played by polars, a notion originating from the theory of lattice ordered groups. One can think of the theory introduced in this section as the analogue to the celebrated Nullstellensatz theorem.
We begin our discussion here considering general idempotent semifields. We will later restrict ourselves to the designated semifield of fractions H(x 1 , ..., x n ) with H a bipotent semifield.
In the following section, S is always assumed to be an idempotent semifield.

Polars
In this subsection we introduce the notion of a polar, borrowed from the theory of lattice ordered groups ( [8, section (2.2)]). We will see that polars are a special kind of kernels and use them to construct the so called K-kernels introduced in the previous section.
Translating Proposition 3.1.9 to the language of idempotent semifields we have: Proposition 6.1.1. The lattice of kernels Con(S) of an idempotent semifield S is complete distributive and it satisfies the infinite distributive law: for any A, B i ∈ Con(S).
Proof. Viewing S as an ℓ-group, since S is commutative every convex ℓ-subgroup is normal and thus a kernel. Thus Proposition 3.1.9 applies to Con(S).
For a ∈ S we shortly write a ⊥ for {a} ⊥ . The set of all polars in S is denoted by B(S).
Remark 6.1.3. The following statements are direct consequences of Definition The following definition generalizes Definition 3.3.1: In particular the positive cone of S is S + = {h ∈ S : h ≥ 1}.
Let G be a group. A subgroup H ≤ G is said to be generated by a family of subgroups {G i : i ∈ I} of G, with an arbitrary index set I, if In particular for every h ∈ H there are some Since the kernel generated by a kernel K (as a set) is the kernel K itself, the last equality holds.
Remark 6.1.8. Let H be an idempotent semifield and let g ∈ H. Then Since the lattice of kernels of S admit the infinite distributive law (6.1) we have that from which the statement follows at once.
Remark 6.1.9. Let K be a kernel of S. The polar of K is the set Proof. For any a, b ∈ S since a ∩ b = |a| ∧ |b| , we have that Moreover, for any kernel K and any x ∈ S, Theorem 6.1.10.

For any subset
2. If L ∈ Con(S), then for any K ∈ Con(S) Closure under negation is a consequence of (2).
Remark 6.1.11. [8] Let A be a subset of the S. Then where G(A) and SF (A) are the subgroup and subsemifield generated by A in S.
Proposition 6.1.13. For any subset S ⊆ S , S ⊥⊥ is the minimal polar containing S.
Proof. By the definition of a polar, S ⊥⊥ is a polar of S ⊥ and S ⊂ S ⊥⊥ . Let P be a polar such that S ⊆ P . Then since polar is inclusion reversing and S ∈ P we have that S ⊥ ⊇ P ⊥ and so S ⊥⊥ = (S ⊥ ) ⊥ ⊆ (P ⊥ ) ⊥ . As P is a polar we have by Remark 6.1.12 that (P ⊥ ) ⊥ = P and thus S ⊥⊥ ⊆ P .
Definition 6.1.14. Let S ⊆ S. We say that a polar P is generated by S if P = S ⊥⊥ . If S = {a} then we also write a ⊥⊥ for the polar generated by {a}.
Remark 6.1.15. The following statements hold: 1. A polar B is generated by itself.

For any subset
where G(A) and SF (A) are the subgroup and subsemifield generated by A in S.
Proof. The first assertion follows directly from Remark 6.1.12 while the second one is a direct consequence of Remark 6.1.11. Definition 6.1.16. A polar P of S is said to be principal if there exists some a ∈ S such that P = a ⊥⊥ , i.e, P is the polar generated by a. We denote the collection of principal polars of a semifield S by PB(S).
Proof. Indeed, by Theorem 6.1.10 and Remark 6.1.11 the following holds: The second assertion is proved analogously.
. The rest of the assertion is a special case of Remark 6.1.18.
Proposition 6.1.20. [8, section (2.2)q-13] The following statements hold for any idempotent semifield S: for any L ∈ Con(K ⊥⊥ ) and K is a large kernel in K ⊥⊥ . Theorem 6.1.22. [8,Theorem (2.3.7)] Consider the following conditions on the kernel K of the idempotent semifield S: 1. K is a polar.

The polar-skeleton correspondence
Recall R is our designated semifield defined to be bipotent, divisible, archimedean and complete. Also recall that the semifield of fractions R(x 1 , ..., x n ) is considered as mapped to the semiring of functions F un(R n , R) as defined in 2.2.33.
Recall that since R is idempotent and archimedean semifield F un(R n , R) is an idempotent and archimedean semifield and so is R( In this section we concentrate our attention to R(x 1 , ..., x n ). Doing so, we consider the natural extensions to R(x 1 , ..., x n ) of the operators Skel and Ker defined in Subsection 4.2 with respect to R(x 1 , ..., x n ). We write Ker R (x 1 , ..., x n ) to denote the restriction of Ker to R(x 1 , ..., x n ).
laws hold for sets with respect to intersections and unions.
Remark 6.2.2. As any kernel K ∈ Con(R(x 1 , ..., x n )) is a bipotent archimedean semifield in its own right, it has a completion too, which we denote byK and K ⊂ R(x 1 , ..., x n ). By Corollary 3.4.10K is a kernel of R(x 1 , ..., x n ).
The question arises which kernels of R(x 1 , ..., x n ) are completions of kernels of R(x 1 , ..., x n ). The answer to this question is all kernels of R(x 1 , ..., x n ) that are completely closed. Indeed, R(x 1 , ..., x n ) is a subsemifield of R(x 1 , ..., x n ), so by Theorem 2.2.51(1) we have that whereL is the completion of L and so, since L ⊆ K, we have thatL =K, i.e., L = K if and only if K is completely closed.
Example 6.2.4. Consider the kernel K = |x| ∧|α| ∈ PCon(R(x)) and the subset Proof. First note that since K ⊂K in R(x 1 , ..., x n ) we have that Skel(K) ⊆ Skel(K). Now, let Z = Skel(K) and let S be any nonempty subset of K. If s∈S s ∈ R(x 1 , ..., x n ) then for any a ∈ Z ( s∈S s)(a) = s∈S s(a) = s∈S 1 = 1, yielding that Skel( s∈S ) ⊇ Z (cf. Remark 3.4.4). Similarly, if s∈S s ∈ R(x 1 , ..., x n ) then for any a ∈ Z we have that ( s∈S s)(a) = s∈S s(a) = s∈S 1 = 1, yielding that Skel( s∈S ) ⊇ Z. Thus as none of its supplementary elements reduces the size of Z, we have that Skel(K) = Skel(K).
Note 6.2.6. Let S be an idempotent semifield and let H be a subset of S. For a subset A of S let We only write A ⊥H when H is a proper subset of the semifield in which A is considered. For example if K ∈ Con(R(x 1 , ..., x n )) then K ⊥ = K ⊥H(x 1 ,...,xn) .
In other words, K is a preimage of its skeleton with respect to the map where P(X) is the powerset of the set X.
., x n )). Let S ⊆ K be any nonempty subset of K then for any s ∈ S we have that Skel(s) ⊇ Z. Now, if s∈S s ∈ R(x 1 , ..., x n ) and s∈S s ∈ R(x 1 , ..., x n ), then by Remark 3.4.4, for any a ∈ Z So s∈S s, s∈S s ∈ K and K is completely closed in R(x 1 , ..., x n ).
Remark 6.2.10. Proposition 6.2.9 is not true when considering K-kernels of So we deduce that K-kernels of R(x 1 , ..., x n ) are not necessarily completely closed in R(x 1 , ..., x n ). Since every polar is completely closed (cf. Theorem 6.1.22), by the example given Proof. By Proposition 6.2.9, every K-kernel of R(x 1 , ..., x n ) is completely closed and thus by Theorem 6.1.22 is a polar since R(x 1 , ..., x n ) is complete.
Summarizing the above assertions we have Corollary 6.2.12. Let K be a kernel of R(x 1 , ..., x n ). Then the following statements are equivalent: 3. K is completely closed. Proposition 6.2.13. For any f ∈ R(x 1 , ..., x n ), Then f ∈ K and K is a polar by Corollary 6.2.11. By Proposition 6.1.13, f ⊥⊥ is the minimal polar containing f thus K ⊇ f ⊥⊥ and so Theorem 6.2.14. There is a 1 : 1 correspondence between the skeletons in R n and the polars of R(x 1 , ..., x n ) given by B → Skel(B) and Z → Ker(Z).
This correspondence restricts to a correspondence between the principal skeletons in R n and the principal polars of R(x 1 , ..., x n ).
By Corollary 6.2.3, the correspondence in Theorem 6.7 yields a correspondence By Corollary 6.2.12 and Remark 6.2.2 for a kernel K ∈ Con(R( By Theorem 6.2.14, (6.6) restricts to a correspondence Thus the K-kernel corresponding to x is the same as the K-kernel corresponding to the (bounded from above) kernel |x| ∧ |α| .
The following example illustrates the necessity of working in R(x 1 , ..., x n ) instead of R(x 1 , ..., x n ). .

Stone representation
As stated in Theorem 6.1.10 for any idempotent semifield S, the collection of polars of S, B(S) forms a complete Boolean algebra with respect to intersection multiplication (corresponding to union) and ⊥ (corresponding to negation). There is a representation for Boolean algebras called the Stone representation.
We will now provide a brief overview of this representation applied to the complete Boolean algebra of polars of S.
The complete construction can be found in [8], (Section (3.2)). Definition 6.3.1. Let X be a topological space. The collection of all clopen (closed which are also open) subsets of X is a Boolean subalgebra of the Boolean algebra of all subsets of X. This collection is called the dual algebra of X. Definition 6.3.2. A compact Hausdorff space whose clopen sets form a base is called a Boolean space. Remark 6.3.3. A closed subset Y of a Boolean space X is also a Boolean space (with respect to the induced subspace topology) and each clopen set in Y is the intersection with Y of a clopen set in X. Definition 6.3.4. A Stone space is a totally disconnected Boolean space which means that every open set is the union of clopen sets. Definition 6.3.5. A Boolean homomorphism between two Boolean algebras is a lattice homomorphism that preserves complements. Definition 6.3.6. The set 2 = {0, 1} is the totally ordered Boolean algebra which will also be considered as a topological space, giving it the discrete topology. Given the Boolean algebra B, the set Bool [B, 2]  • Each Boolean algebra B is isomorphic to its second dual, i.e., the dual algebra B of the Boolean space Bool [B, 2]. The isomorphism α : B → B is given by • Each Boolean space is isomorphic to its second dual. Namely, let A be the dual algebra of a Boolean space X, and let Y = Bool[A, 2] be the dual space of A. Then the function β : X → Y given by In view of the above, Spec(B(S)) is a Stone space. subset of X.
Remark 6.3.9. E(X) is a poset with respect to the coordinate-wise partial order: for each x ∈ X, and, in fact, E(X) is a sublattice of the for each x in some dense subset of X. The subset D(X) is a sublattice of E(X) since the intersection of two dense open subsets is dense.
Theorem 6.3.11. [8,Theorem (2.3.23)] Let S be an archimedean idempotent semifield, and let X = Spec(B(S)) be the Stone space of the Boolean algebra B(S) of polars of S. Then there is a (complete) cl-essential monomorphism from S into D(X).

The principal bounded kernel -principal skeleton correspondence
Recall that the semifield R is defined to be a bipotent divisible archimedean complete semifield. By Corollary 3.4.14 one can regard R as being (R + , ∔, ·).
Throughout this section, H is assumed to be a bipotent divisible semifield. Any supplementary assumptions on H will be explicitly stated.
Although we have already established a correspondence between skeletons and polars which restricts to a correspondence between principal skeletons and principal polars, we proceed to find a correspondence between principal skeletons to principal kernels of a very special kernel of R(x 1 , ..., x n ), namely, the kernel R presented above which will now serve us as a semifield of its own right. It turns out that R possesses just enough distinct bounded copies of the principal kernels of R(x 1 , ..., x n ) to represent the principal skeleton without any ambiguity.
. Then all the statements introduced in the section 'Skeletons and kernels of skeletons' apply to Ker R and Skel| R .
Note 7.0.13. As Ker R and Skel| R are our actual object of interest, we also denote them by Ker and Skel. The distinction will be carried out through the context of discussion. If an ambiguity arises we will explicitly say to which Ker or Skel we refer.
We now show that restricting Skel and Ker does not affect the collection of resulting skeletons and that each K-kernel of R(x 1 , ..., x n ) has a K-kernel (with respect to the restriction) in R .
where X is any set generating K as a kernel, in particular, one can take X = K. Now, for any f ∈ R(x 1 , ..., x n ), in particular in Note that these restricted kernels are exactly the kernels of the (domain) restric- By Proposition 4.2.5 and the above discussion we have Proposition 7.0.15. There is a 1 : 1 order reversing correspondence

2)
given by Z → Ker(Z) ∩ R ; the reverse map is given by K → Skel(K) with K ∈ Con( R ).
Let S be an idempotent semifield. Recall that the positive cone of S is By Theorems 3.3.2 and 3.3.4 (applied to idempotent semifields), S + is a subsemigroup of S which is a lattice (i.e., closed with respect to ∔ (i.e., ∨) and ∧.
Let f ∈ H(x 1 , ..., x n ) and fix α = 1 in H. Then Note that for any f ∈ H(x 1 , ..., x n ), |f | is a generator of f .
We define the map Then ω is a lattice homomorphism. Indeed, and Let us study the map Ω and its image. 2. If f is a bounded from below kernel, then f ∩ H = H by Remark 5.1.13. In fact, by the above discussion of bounded from below functions, we see that every principal kernel whose skeleton is the empty set is mapped to H .
3. As Skel( α ) = ∅ and since Skel( f ∩ g ) = Skel( f ) ∪ Skel( g ), for any principal kernel f we have that Thus Ω does not affect the skeleton of a kernel, the skeleton is fixed.
As we shall shortly show, there exists a correspondence between the principal skeletons in R n and the kernels in PCon( R ). If Skel(g) = Skel(f ) then since Skel(g) = Skel(Ω(g)) we have that Skel(Ω(g)) = Skel(f ) ∈ PCon ( R ). Thus in view of the above Ω( g ) = f . Consequently we have that Skel(K) = Skel(f ) and K is the maximal kernel of R(x 1 , ..., x n ) having this property.
The choice of R for our algebraic infrastructure is a natural choice, as we have R as our basic semifield, with respect to which homomorphisms are taken.
Taking the semifield in R(x 1 , ..., x n ) generated by R as a kernel, we get the only semifield homomorphism R → H (where H is a semifield containing R) sending α to 1 is the trivial one R → {1}. All proper subkernels K of R admit K ∩ R = {1} which is a necessary condition for a kernel of an R-homomorphism (which is otherwise not well defined, as any α = 1 is required to be mapped to itself). One can view the elements of R \ {1} as playing the role of invertible elements in rings, in the sense that any ideal containing an invertible element is the ring itself.
Note 7.0.17. Since the principal kernels in PCon( R ) are in particular principal kernels in PCon(R(x 1 , ..., x n )), in the remainder of this paper we generally study PCon(R(x 1 , ..., x n )) where the results are true in particular for PCon ( R ). In our subsequent discussions we develop the notion of reducibility, regularity and corner-integrality for a principal kernel in PCon(R(x 1 , ..., x n )). We develop the theory of reducibility for general sublattices of PCon(R(x 1 , ..., x n )), thus in particular for sublattices of PCon ( R ). In the sections to follow, we introduce the notions of regularity and corner-integrality of kernels which are oriented to skeletons, in the sense that we consider points for which a generator f (equivalently |f |) of the kernel attains the value 1. Since passing to |f | ∧ |α| for some α = 1 in R does not affect these points, these notions stay intact restricting them to PCon ( R ), in the sense that |f | is regular (corner-integral) if and only if |f | ∧ |α| is regular (corner-integral). In those places where it is necessary, we explicitly restrict ourselves to PCon( R ). Proof. Although the first statement was proved in Proposition 4.1.5(2), we present here a somewhat more elegant proof. By Corollary 2.3.14 we have that g ∈ R(x 1 , ..., x n ) is in f if and only if there exists some n ∈ N such that (f + f −1 ) −n ≤ g ≤ (f + f −1 ) n , or by different notation, |f | −n ≤ g ≤ |f | n . Now, since over Skel(f ), f = f −1 = 1, we have 1 = 1 −n ≤ g ≤ 1 n = 1 over Skel(f ) and thus g = 1 for every g ∈ f .
For the second assertion, let f ′ be a generator of f then by Corollary 2.3.14 for some k ∈ N, |f ′ | −k ≤ f ≤ |f ′ | n , which yields, using the first statement that for Proof. First, note that by Proposition 4.1.5(1), we have that Skel(f ) ⊆ Skel(h) since h ⊆ f . The 'only if' part of the assertion follows from the fact that if h is a generator then h = f and thus their skeletons coincide.
Note 7.0.20. In the following proposition we use the property of R being complete, in the sense that the underlying lattice of R is conditionally complete (see Definition 2.2.27).  (3) Before we continue, note that R is a totally ordered semifield, thus for any Let h ∈ f such that h is not a generator of f while Skel(h) = Skel(f ). By Corollary 2.3.14 we have that for each k ∈ N there exists some x k ∈ R n for which |f (x k )| > |h(x k )| k (*). Note that f and h are rational polynomials thus continuous and so are |f | s and |h| s (by definition of | · |) for any s ∈ N. For any k ∈ N, define the set U k = {x : |f (x)| > |h(x)| k }. As R is assumed to be (ordered) divisibly closed semifield, it is dense, so, for any x ∈ U k there exists a neighborhood B x ⊂ U k containing x such that for all x ′ ∈ B x , |f (x ′ )| > |h(x ′ )| k . Now, since h and f are bounded from above rational functions both not equal to 1, U k are bounded regions inside R n . Taking the closure of U k we may assume it is closed. Since where by our assumption (*), U i = ∅. Thus, since R is complete, there exists an element y ∈ R such that y ∈ B = N B k . Now, for x ∈ Skel(h), |h(x)| > 1 thus there exists some r = r(x) ∈ N such that |h(x)| r > |f (x)| thus x ∈ B thus y ∈ R n \ Skel(h) . On the other hand, if y ∈ Skel(h) then y ∈ Skel(f ) so 1 = |f (y)| ≤ |h(y)| = 1. Thus N B k = ∅. A contradiction.
Note 7.0.22. In the last proof we could analogously argue that since y ∈ Skel(h), f (y) > h(y) k for every natural number k where h(y) > 1, which yields that f (y) ∈ R = h(y) .
If h ∈ f is such that Skel(h) = Skel(f ) and |h| has an essential expansion as |h| = k i=1 s i |f | d(i) with d(i) ∈ Z and s 1 , ..., s k ∈ R(x 1 , ..., x n ) such that k i=1 s i = 1, then h is a generator of f .
Proof. As |f | is a generator of f and |h| a generator of h , we may consider |f | and |h| instead of f and h and moreover, we may assume |f | and |h| to be in essential form. By Proposition 2.3.8, there exist some s 1 , ..., s k ∈ R(x 1 , ..., x n ) such for every y ∈ ǫ. But k i=1 s i = 1 thus s i 0 (y) = 1 for every y ∈ ǫ. Without loss of generality, take i 0 = 1. If ǫ ⊆ Skel(f ) then, as R is divisibly closed, there exists a point y 1 ∈ ǫ \ Skel(f ) such that |h(y 1 )| = s 1 (y 1 ) + k 2 s i (y 1 )|f (y 1 )| d(i) = s 1 (y 1 ) = 1, which contradicts the assumption that Skel(h) = Skel(f ). So ǫ ⊆ Skel(f ).
for every x ∈ R (since ǫ ∩ R = ∅ there always exists some 2 ≤ j ≤ k for which s j (x) = 1). Now, take x ∈ R. Then there exists some 2 ≤ j 0 ≤ k such that s j 0 (x) = 1, thus we have that |h( , |h| ≥ |f |. Thus, by Remark 2.3.16 |f | ∈ |h| , so f = |f | ⊆ |h| = h . Finally, as h ∈ f , we have that h = f , i.e., h is a generator of f as desired. The proof of Proposition 7.0.23 does not apply to any element of f , just to the element which can be written essentially in the form a = k i=1 s i |f | d(i) with k i=1 s i = 1. For example, the element |x| ∧ α ∈ x with α > 1 can be written as ( |x| α+|x| ) · 1 + ( α α+|x| ) · |x| with a 1 (x) = |x| α+|x| and a 2 (x) = α α+|x| , but this is not in an essential form since the first term never dominates. Proof. We need to show that (Ker(Skel(f )) =) Ker(Skel( f )) ⊆ f .
Let h ∈ R such that h ∈ Ker(Skel(f )). Then h(x) = 1 for every x ∈ Skel(f ) and so Skel(f ) ⊆ Skel(h). If |h| ≤ |f | k for some k ∈ N then h ∈ f . Thus in particular we may assume that h = 1. Now, by Corollary 4.2.7 we have that have that g ∈ f and Skel(g) = Skel(h). Thus By Proposition 7.0.21, g is a generator of h , so, we have that h = K ⊆ f as desired. given by Z → Ker R (Z); the reverse map is given by K → Skel(K).
Proof. Every principal kernel gives rise to a principal skeleton by the definition of Skel. The reverse direction follows Proposition 7.0.21 as every principal kernel which produces a principal skeleton using Skel is in fact a K-kernel.
In Proposition 2.5.15, we have shown using a substitution homomorphism ψ that any point a = (α 1 , ..., α n ) ∈ R n corresponds to the maximal kernel We will now show that any maximal kernel of R is of that form.

The coordinate semifield of a skeleton
In this section we define the coordinate semifield corresponding to a skeleton.
Being the most relevant to the development achieved in the reminder of this work, we first perform the construction for principal skeletons, using the principal kernels of R . Later on we introduce a more general construction of the coordinate semifield of a (principal) skeleton using its corresponding (principal) polar.
to be the restriction map f → f | V . Then φ V is a homomorphism and Proof. For any g, h ∈ R(x 1 , ..., x n ), since φ V is a restriction map, we have that , as desired.
as groups.
Proof. Let f 1 and f 2 be the kernels in PCon( R ) such that V 1 = Skel( f 1 ) and The following result is analogue to Proposition 8.0.29 using R(x 1 , ..., x n ) and B defined in Corollary 6.2.15 instead of R and Con( R ), respectively.
to be the restriction map f → f | V . Then φ V is a homomorphism and where K S = S ⊥⊥ ∩R(x 1 , ..., x n ) with S ⊥⊥ is taken in the completion R(x 1 , ..., x n ) of R(x 1 , ..., x n ) in F un (R n , R). In the previous sections we have proved some correspondences between skeletons and kernels. In particular we have shown a correspondence between principal skeletons and principal kernels. We now turn to find the connection between tropical varieties (which we call corner loci) and skeletons.
Before diving into the core of the theory we develop some tools and notion to facilitate our construction.
Throughout this section, H is assumed to be a bipotent divisible semifield. Any supplementary assumptions on H will be explicitly stated. We also recall that our designated semifield R is defined to be bipotent, divisible, archimedean and complete the prototype being (R + , ∔, ·) by Corollary 3.4.14.

Essentiality of elements in the semifield of fractions
In the theory of tropical geometry, there exists a notion of essentiality of monomials in a given polynomial.
i.e., m 1 never solely dominates p over D.
Note 9.1.2. In the following section, we introduce a notion of essentiality for elements of H(x 1 , ..., x n ). This notion differs from the tropical one, in fact it generalizes it in some sense, as we will show shortly. Generally, the context will imply the relevant notion among the two. In case ambiguity arises, we will explicitly indicate the one we refer to.
Proof. This statement is a direct consequence of the definitions.
Definition 9.1.5. Let f, g ∈ H(x 1 , ..., x n ). g is said to be inessential for f if Otherwise f j is essential in f . Note 9.1.6. Note that inessentiality defined in Definition 9.1.5 differs from the notion of inessentiality in tropical geometry. In tropical geometry, a monomial of a polynomial is considered inessential if it is not dominant anywhere, in the sense that it does not attain solely the maximal value of the polynomial.
., x n ), the following condition holds: In words each f i is essential in f .
Remark 9.1.9. Note that in order for a monomial h i to be essential, there is no need for the occurrence of some g j such that h i (x) = g j (x) for some x ∈ H n and viceversa. The reason for this is that h i can affect the skeleton by preventing another monomial h ′ of the numerator to dominate h in a point x ∈ H n where h ′ (x) = g(x). The inessential monomials h i and g j in f = h g are characterized as follows: A monomial h ′ of h is inessential in f if one of the following two conditions holds for all x ∈ H n : Analogously, a monomial g ′ of g is inessential in f if one of the following two conditions holds for all x ∈ H n : It can easily be seen that h ′ and g ′ admitting the above criterion do not affect the skeleton of f . Moreover, if a monomial g ′ is inessential in g then takingf = h g whereg is defined to be g with g ′ omitted, we have that any monomial g ′′ = g ′ of g and any monomial h ′ of h are essential inf if and only if they are essential in f . In view of the above, we can define f e ∈ H(x 1 , ..., x n ) to be the rational function obtained from f = g h by omitting all inessential monomials in f . By the above, f e is well-defined regardless of the order with respect to which the monomials are omitted.
In view of the above discussion the following observations hold:  Proof. Indeed, if h is inessential one of the composing monomials of h, say h ′ , does not affect the values h obtains and thus does not affect the skeleton of f and the summand h ′ g can be omitted from f without changing Skel(f ). If g is inessential, then since Skel(f ) = Skel(f −1 ) we can consider g h what brings us back to the previous case considered. Namely, if g ′ is an inessential monomial of g, g ′ h can be omitted from g h . Now, taking the inverse of the resulting fraction brings us back to f with the monomial g ′ omitted.

Reducibility of principal kernels and skeletons
In this section we consider the notion of reducibility with respect to a sublattice of kernels of the semifield H(x 1 , ..., x n ). The sublattice of kernels that is of interest to us are actually contained inside the lattice PCon ( R ). Note that PCon ( R ) is both a sublattice of Con( R ) and of Con(R(x 1 , ..., x n )). Definition 9.2.1. Let S be a semifield. A subset Θ of Con(S) is said to be a sublattice of kernels if for every pair of kernels K 1 , K 2 ∈ Θ, Example 9.2.2. Con(R(x 1 , ..., x n )), PCon(R(x 1 , ..., x n )) are sublattices of kernels of R(x 1 , ..., x n ). Con ( R ), PCon( R ) are sublattices of kernels of R(x 1 , ..., x n ) and of R .
Definition 9.2.3. Let Θ be a sublattice of kernels of a semifield S. A proper (nontrivial) kernel K ∈ Θ is called Θ-irreducible if for any pair of kernels A, B ∈ Θ Note that if a kernel K ∈ Θ is Θ-irreducible then K is weakly Θ-irreducible.
Definition 9.2.4. Let S be a semifield and let Θ be a sublattice of kernels. Then S is said to be Θ-irreducible if for any pair of kernels K 1 ∈ Θ and K 2 ∈ Θ such Remark 9.2.5. If K is an Θ-irreducible kernel of S, then the quotient semifield U = S/K is Θ-irreducible.
Note 9.2.6. In particular, taking S to be an idempotent semifield one can take θ to be the sublattice PCon(S), i.e., restrict the notion of reducibility to the principal kernels of S.
Having definitions for reducibility and irreducibility of (principal) kernels, we now turn to define the analogous geometric notion for principal skeletons (for the case S = H(x 1 , ..., x n )).
Definition 9.2.7. Let Θ be a sublattice of kernels in H(x 1 , ..., x n ). A skeleton S is said to be a Θ-skeleton if there exists some kernel K ∈ Θ such that S = Skel(K).
Definition 9.2.8. Let Θ be a sublattice of kernels in H(x 1 , ..., x n ). A Θ-skeleton S is said to be Θ-reducible if there exist some Θ-skeletons S 1 and S 2 such that S = S 1 ∪ S 2 and S = S 1 and S = S 2 ; otherwise S is Θ-irreducible. Let Θ ⊆ PCon(H(x 1 , ..., x n )) be a sublattice of kernels. A principal (finitely generated) skeleton Skel(f ) is said to be Θ-reducible if there exist principal Θ-skeletons Skel(g) and Skel(h) such that Skel(f ) = Skel(g) ∪ Skel (h) and Remark 9.2.9. By Definition 9.2.8, Skel(f ) is Θ-irreducible if for any pair of Θ-skeletons Skel(g) and Skel(h) such that Skel(f ) = Skel(g) ∪ Skel(h), either Skel(f ) = Skel(g) or Skel(f ) = Skel(h). Translating this last statement to the kernels f , g and h in P con ( R ), and using the principal kernels -principal skeletons correspondence, we get the condition stated in Definition 9.2.3 of Θ-irreducible kernels.
By the last remark we have We now turn to study more closely the notion of reducibility of a kernel, and introduce adequate geometric interpretations for reducibility of its skeleton. We begin our discussion by introducing the notion of a reducible element of R(x 1 , ..., x n ) corresponding to reducibility of the principal kernel it defines.
In the theory of commutative rings a generator of a principal ideal is unique up to multiplication by an invertible element, i.e., the association class of a generator of an ideal is unique. Recall that for two elements a and b of a commutative ring R, a and b are associates if and only if a|b and b|a. In our setting things are slightly more complicated. We consider ∼ K equivalence classes of generators of kernels. The equivalence ∼ K will be shown to be induced by a certain order relation, , defined on the elements of the semifield and plays analogous role to that of |.
Definition 9.2.11. Let S be a semifield and let Θ be a sublattice of kernels in S. An element a ∈ S is said to be a Θ-element if a ∈ Θ.
Remark 9.2.12. Following Definition 9.2.11, any generator b ∈ S of a is a Θ-element.
Notation 9.2.14. Throughout the rest of this subsection we continue taking Θ ⊆ Con(H(x 1 , ..., x n )) to be a sublattice of kernels.
Definition 9.2.15. Let S be a semifield and let a, b ∈ S. Define the following relation on S This is clearly an equivalence relation, the classes of which are [a] = {a ′ : a ′ is a generator of a }.
Define the partial relation on S as follows: Definition 9.2.16. Let S be a semifield and let a, b ∈ S. We say that a and b are k-comparable if a b or b a, i.e., if there exist some a ′ ∼ K a and b ′ ∼ K b such that |a ′ | and |b ′ | are comparable |a ′ | ≤ |b ′ | or |b ′ | ≤ |a ′ |.
We introduce explicitly the translation of Definition 9.2.15 for the case where the semifield S is H(x 1 , ..., x n ) with H a bipotent divisible semifield.
Remark 9.2.17. For every h, g ∈ H(x 1 , ..., x n ) such that g, h ≥ 1, Proof. If g ≥ h then taking w = h ≥ 1 we have h = g ∧h. Conversely, if h = g ∧w then g ≥ g ∧ w = h. Moreover w must admit w ≥ 1 for otherwise if w(a) < 1 for some a ∈ H then h(a) = g(a) ∧ w(a) ≤ w(a) < 1 contradicting the assumption that h ≥ 1.
Since h ∼ K |h| ≥ 1 for any h ∈ H(x 1 , ..., x n ), by Remark 9.2.17 the relation on H(x 1 , ..., x n ) can be stated as follows: Remark 9.2.19. In view of Corollary 2.3.14 the partial relation (9.9) of Definition 9.2.18 can be rephrased as Indeed, if f ′ ∈ [f ] then there exists k ∈ N such that |f ′ | ≤ |f | k , and thus |f ′ | = |f | k ∧ v for some v ∈ H(x 1 , ..., x n ) where v ≥ 1 . Assume f g, then, by (9.9), since v, w ≥ 1 we have that v ∧ w ≥ 1 which means that (9.10) holds for g. The converse direction is obvious, since |f | k ∈ [f ] and |f | k ≥ 1 for any k ∈ N. Proposition 9.2.20. For any f, g ∈ H(x 1 , ..., x n ), g f ⇔ g ⊇ f . (9.11) Proof. Let f, g ∈ H(x 1 , ..., x n ) such that g f . Then, by (9.10) we have that g f if and only if f ∼ K |g| k ∧ w for some w ≥ 1 and k ∈ N, if and only if there exists some f ′ ∼ f such that f ′ = |g| k ∧ w. Note that |g| k ≥ 1 and w ≥ 1

w ⊂ f (with strict inclusion).
Proof. By condition (9.10) and Proposition 9.2.20. Proof. f ∼ K g ⇔ f = g , which, by Proposition 9.2.20 is equivalent to f g and g f .
As stated above, the set of principal kernels of R(x 1 , ..., x n ), PCon(H(x 1 , ..., x n )), forms a lattice with respect to multiplication and intersection. Irreducibility of a kernel is translated there as follows: otherwise, it is Θ-reducible. When Θ = PCon(H(x 1 , ..., x n )) we simply say reducible (irreducible).  (1) holds for H(x 1 , ..., x n ) implying that the following condition is equivalent to condition 9.15: Remark 9.2.28. By transitivity of , if f 1 g 1 , f 1 ∼ K f 2 and g 1 ∼ K g 2 , then f 2 g 2 Remark 9.2.29. For Θ-elements f and f ′ , such that and only if f f ′ and f ′ f . Assume f is Θ-irreducible. If g and h are Θ-elements such that g ∧ h f ′ , then g f ′ and h f ′ . Since f ′ f we get that g f and h f . By Θ-irreducibility of f we have that f g or f h. Now, as f ′ f we get that f ′ g or f ′ h as desired. The arguments of the proof in the opposite direction of the assertion are symmetric.
Note 9.2.30. Remark 9.2.29 actually follows irreducibility being defined by (while respects ∼ K ). Nevertheless, we prove it explicitly.
The following Proposition establishes the connection between irreducible principal kernels and irreducible elements of H(x 1 , ..., x n ). Proof. If f ∼ K |g| ∧ |h| then f = |g| ∧ |h| = g ∩ h . Since f is irreducible we have that either g ⊆ f or h ⊆ f , which is equivalent, by Proposition 9.2.20 to f |g| or f |h|. Now, as |g| |g| ∧ |h| and |h| |g| ∧ |h| we get that Then f = |g| ∧ |h| , and thus f ∼ K |g| ∧ |h|. As f is irreducible, we have that f ∼ K |g| or f ∼ K |h| and so f = |g| = g or f = |h| = h as desired.
The proof of the following Corollary is completely analogous.
Remark 9.2.34. f ∈ H(x 1 , ..., x n ) is reducible if and only if there exist g, h ∈ H(x 1 , ..., x n ), g, h ≤ 1, such that f ∼ K g ∔ h where f ∼ K g and f ∼ K h.
Equivalently, f is irreducible if for any such g and h, f ∼ K g or f ∼ K h Proof. The definition of ∧ implies that g −1 ∧ h −1 = (g ∔ h) −1 ∼ K g + h. Now, since s −1 ∼ K s for every s ∈ H, by definition of irreducibility we have that f ∼ K g −1 or f ∼ K h−1 if and only if f ∼ K g or f ∼ K h which yields the stated conclusion.
Lemma 9.2.35. Let f ∈ H(x 1 , ..., x n ) be a rational function. We can write if and only if for every 1 ≤ i ≤ k the following condition holds: x ∈ H n such that f (x) = 1, then there exists some i ∈ {1, ..., n} such that f i (x) = 1 and f j (x) ≤ 1 for every j = i. Thus |f i |(x) = |f i (x)| = 1 and |f j |(x) = |f j (x)| ≥ 1, yielding that Conversely, iff (x) = 1 then there exists some i ∈ {1, ..., n} such that |f i (x)| = 1 and |f j (x)| ≥ 1 for every j = i. Now, |f i (x)| = 1 if and only if f i (x) = 1 and as condition (9.17) holds, we get that f j (x) ≤ 1 for all j = i. Thus

Decompositions
., x n ] and g i is a monomial. If each time the value 1 is attained by one of the terms f i in this expansion and all other terms attain values smaller or equal to 1, thenf = k i=1 |f i | defines the same skeleton as f . Moreover, if f ∈ H thenf ∧ |α| ∈ H , for α ∈ H \ {1} is also a generator of f . The reason we takef ∧ |α| is that we have no guarantee that each of the f i 's in the above expansion is bounded.
We can generalize this idea as follows: ., x n ], (h being the common denominator derived from the h i 's above), such that for every 1 ≤ i ≤ k the following condition holds: The decomposition (9.18 18), is said to be non-trivial. A decomposition (9.18) is said to be a Θ-decomposition if both u and v (equivalently |u| and |v|) are Θ-elements (and thus, so is f ) . Proof. If f is Θ-reducible then there exist some kernels u and v in Θ such we have that f ′ = |u| ∧ |v| is a generator of f which by the above is a nontrivial Since f ′ = |u| ∧ |v| is nontrivial, we have that u ∼ K f ′ and v ∼ K f ′ thus |u| = u = f ′ = f and similarly v = f . Thus, by definition, f is Θ-reducible.
We can equivalently rephrase Lemma 9.3.2 as follows: There is an immediate question arising from Definition 9.3.1 and Lemma 9.3.2: If f ∈ H(x 1 , ..., x n ) has a non-trivial Θ-decomposition and g ∼ K f , does g have a non-trivial Θ-decomposition too, and if so, what is the relation between this pair of decompositions?
In the following few paragraphs we will give an answer to both of these questions in the case θ = PCon( R ).
Proof. If f is a principal reducible kernel, then there exists some f ′ ∼ K f such that f ′ = |u| ∧ |v| = min(|u|, |v|) for some Θ-elements u, v ∈ R where f ′ ∼ K |u|, |v|. Since f ′ is a generator of f , we have that |f | ∈ f ′ , so there exists some s 1 , ..., s k ∈ H(x 1 , ..., x n ) such that k i=1 s i = 1 and |f | i) . Now, by the above we have that |f | ⊆ |g| ⊆ |u| and |f | ⊆ |h| ⊆ |v| , thus Skel(f ) ⊇ Skel(g) ⊇ Skel(u) and Skel(f ) ⊇ Skel(h) ⊇ Skel(v). We claim that |g| and |h| generate |u| and |v| , respectively. Since f ′ ∼ K |f | we have that Skel(f ′ ) = Skel(|f |), Then x ∈ Skel(g). Then we have that g(x) = 1, i.e., k i=1 s i |u| d(i) = 1. Let s t |u| d(t) be a dominant term of g at x. If s t (x) = 1 then |u| d(t) = 1 and thus u = 1 and x ∈ Skel(u) otherwise s t (x) < 1 (since k i=1 s i = 1) and so, by the above s t (f ′ ) d(t) is not a dominant term of |f | at x. Since s t (x) < 1 we have that for any dominant term of |f | at x. Thus On the other hand, as Skel(f ) ⊇ Skel(g), we have that f ′ (x) = 1 and thus s j (x)(f ′ (x)) d(j) = 1, contradicting (9.19). So, we have that Skel(g) ⊆ Skel(u), so, by the above Skel(g) = Skel(u) which in turn yields that g is a generator of |u| = u . The proof for h and |v| is analogous. Consequently, we have that g ∼ K |g| ∼ K |u| and h ∼ K |h| ∼ K |v|, so, as |f | ∼ K f ′ ∼ K |u|, |v| we have that |f | ∼ K |g|, |h|.
Proof. Follows successive application of Theorem 9.3.6.
Remark 9.3.8. Let H be a bipotent divisible semifield. Let f ∈ H(x 1 , ..., x n ), and let g ∈ f such that g ∼ K f . As g ∈ f , g can be written as a convex expansion of g with respect to f . Then for any x ∈ H n , g(x) = 1 if and only if s j (x) = 1 for any leading term s j f d(j) of g at x.
Corollary 9.3.9. If f is a kernel in Θ, then f has a nontrivial Θ-decomposition f = g ∩ h if and only if |f | has a non-trivial decomposition |f | = |g ′ | ∧ |h ′ | with |g ′ | ∼ K g and |h ′ | ∼ K h.
Conversely, if g and h are not k-comparable then g h and h g, so by Proposition 9.2.20, h ⊆ g and g ⊆ h respectively. Then f = g ∩ h = g , h , which yields that f ∼ K g and f ∼ K h, and so f is Θ-reducible.

Regularity
Recall that our designated semifield R is defined to be bipotent, divisible, archimedean and complete, the prototype being (R + , ∔, ·) by Corollary 3.4.14.
Thus R ( respectively R n ) can be thought of as a topological metric subspace of R (respectively R n ) with the induced topology derived from the usual (Euclidian) topology of R (respectively R n ).
There are two general types of nontrivial principal skeletons in R n : Skeletons not containing a region of dimension n and skeletons that do contain a region of dimension n. These two types of principal skeletons emerge from two distinct types of kernels, characterized by their generators. The first type of skeletons correspond to principal kernels generated by an element of R(x 1 , ..., x n ) which we call regular while the second type of skeletons correspond to principal kernels generated by an irregular element of R(x 1 , ..., x n ). Principal kernels encapsulate a relation of the form f = 1 for some f ∈ R(x 1 , ..., x n ), which is induced on the quotient semifield. The relation f = g i h j = 1 is local by nature in the following sense: Let x ∈ R n be any point. Then there is at least one monomial f i0 of the numerator and at least one monomial g j0 of the denominator which are dominant at x. If more than one monomial is dominant in each case, say {f ik } s k=1 and {g jm } t m=1 , then we have some set of additional relations of the form f i0 = f ik and g j0 = g jm . Now, the non-dominant monomials of both numerator and denominator define some order relations on the variables, which in turn define a region of R n over which the relations {f ik = g jm : 0 ≤ k ≤ s, 0 ≤ m ≤ t} hold. Every such relation translates by multiplying by inverses of variables to a relation of the form 1 = φ(x 1 , ..., x n ) with φ ∈ R(x 1 , ..., x n ) \ R a Laurent monomial, and thus reduces the dimension. Note that in the special case in which f i0 and g j0 singly dominate and are the same monomial, no relation is imposed on the region described above, so we are left only with order relations defining the region. In essence, leaving only dominant monomials at a neighborhood of a point, such local relations fall into two distinct cases: • An order relation of the form 1 ∔ g = 1 with g ∈ R(x 1 , ..., x n ), which in fact describes a relation of the form s ∔ t = s with s, t ∈ R[x 1 , ..., x n ]. The resulting quotient semifield R(x 1 , ..., x n )/ 1 ∔ g does not reduce the dimension of R(x 1 , ..., x n ), but only imposes new order relations on the variables.
• A 'regular' relation, in the sense that it is not an order relation. Such a relation reduces the dimensionality of the image of R(x 1 , ..., x n ) in the quotient semifield.
Both of these categories are illustrated below and will be studied in Section 13 concerning dimensionality of kernels and skeletons. We define a regular element of R(x 1 , ..., x n ) to be such that does not translate (locally) to order relations but only to regular relations (locally). This observation will allow us to characterize those relations which correspond to corner loci (tropical varieties in tropical geometry). The kernels corresponding to these relations will be shown to form a sublattice of the lattice of principal kernels (which is itself a sublattice of the lattice of kernels).
In the following, we will characterize the generators of principal kernels of R(x 1 , ..., x n ) which correspond to corner loci, which we call corner integral rational functions. In this subset of elements of R(x 1 , ..., x n ), the regular elements correspond to the traditional tropical varieties considered in tropical geometry, which are precisely the supertropical varieties defined by tangible polynomials (see subsection 10 and [4]), while the irregular correspond to supertropical varieties defined by supertropical nontangible polynomials. Evidently 'tangible' polynomials form a multiplicative subset in the domain of supertropical polynomials.
Example 9.4.1. Consider the quotient map φ : R(x) → R(x)/ x ∔ 1 . This map imposes the relation x ∔ 1 = 1 on R(x), which is just the order relation x and1 are comparable as opposed to the situation in R, where x and 1 are not comparable, i.e., do not admit any order relation. If instead of considering x ∔ 1 we consider |x| ∔ 1 = x ∔ x −1 ∔ 1, then as |x| ≥ 1 the relation |x| ∔ 1 = 1 is in fact |x| = 1 which yields the substitution map sending x to 1. Note that |x| and 1 are comparable in R(x), as mentioned above, |x| ≥ 1, which is equivalent to the relation imposed by the equality |x| ∔ 1 = |x| or equivalently by |x| −1 ∔ 1 = 1. The kernel |x| −1 ∔ 1 , as |x| −1 ∔ 1 = 1, is just the trivial kernel 1 = {1}.
As seen in Proposition 2.3.11 and Corollary 2.3.14, order relations affect the structures of kernels in a semifield. For instance, consider a principal kernel in a semifield R generated by an element a ∈ R. Then b ∈ a for any element b ∈ R such that b is not comparable to a. Definition 9.4.2. Let S be a bipotent semifield. Let f = h g = k i=1 h i m j=1 g j ∈ S(x 1 , ..., x n ) such that h i and g j are monomials for all 1 ≤ i ≤ k and 1 ≤ j ≤ m.
Then f is said to be regular if for each x ∈ S n , there exist h i and g j such that h(x) = h i (x), g(x) = g j (x), and h i = g j . Otherwise f is called irregular. Notation 9.4.3. Let Reg(R(x 1 , ..., x n )) denote the set of regular elements in R(x 1 , ..., x n ).

Proof. f −1 is regular since the definition of regularity is invariant taking inverses.
f k is regular follows easily from the regularity of f . We will now prove that f ∔ g is regular. Let f, g ∈ Reg(R(x 1 , ..., x n )).
Then in particular s j dominates s at x and since f = 1 there exists some Again, as noted above we have that t i w j = s i ′ w j as desired. Finally, f ∔ f −1 and f ∧ g = (f −1 ∔ g −1 ) −1 which yields that they are regular.
The Skeleton Skel(f ) corresponding to K is said to be regular also. We denote the family of regular skeletons by RegSkl(S n ) and the family regular principal skeletons by RegP Skl(S n ).
We need to show that regularity of a principal kernel is well-defined, i.e., that it is independent of the choice of the generator of the kernel.
Remark 9.4.6. Let f ′ ∈ f be a generator of f . Then as we have previously shown Assume that f ′ is irregular. Then there exists some a ∈ R n such that h(a) = h i 0 (a) = g j 0 (a) = g(a) and h i 0 = g j 0 where for every i = i 0 and j = j 0 h i (a) < h i 0 (a) and g j (a) < g j 0 (a), respectively. Now, since f ′ is continuous and by the definition of R (being isomorphic to R + ) there is a neighborhood ε(a) of a, {a} ⊂ ε(a) ⊆ R n for which the above holds implying that Skel(f ′ ) = Skel(f ) since by the definition of regularity no such neighborhood exists.
1 is not regular since 1 being essential in the numerator coincides with the denominator over some nonempty region. Note 9.4.8. Note that there may exist a mutual monomial of h and g, but in such a case, it can not dominate both h and g at the same point. Corollary 9.4.9. The set of regular principal kernels forms a sublattice of PCon (R(x 1 , ..., x n )).

Corner-Integrality
In other words, f ∈ S(x 1 , ..., x n ) is corner integral if for any x ∈ S n , if x is a corner root of h then g(x) ≥ h(x) (i.e., g surpasses h at x) and if x is a corner root of g then h(x) ≥ g(x) (i.e., h surpasses g at x).
In other words, a principal kernel is corner-integral if it has a corner-integral generator. The skeleton Skel(K) corresponding to K is said to be a cornerintegral skeleton.
Remark 9.5.3. When f ∈ S(x 1 , ..., x n ) is said to be corner integral we always take f to be a reduced fraction. For example, x ∈ R(x) is trivially corner-integral, though (x+α)x x+α for α > 1 is not -since substituting α for x we get α 2 +α 2 α+α . Thus α is a corner root of the numerator which is not surpassed by the denominator since α 2 > α. Notation 9.5.4. We denote by CI (S(x 1 , ..., x n )) the set of corner-integral elements of S(x 1 , ..., x n ).
Remark 9.5.5. Let f ∈ R(x 1 , ..., x n ) be corner-integral. Then f −1 , f k , for any k ∈ N, and m i=1 f d(i) with d(i) ∈ Z, also are corner integrals.
Proof. First, f −1 is corner integral as the definition of corner integrality is invariant w.r.t taking inverses.
., x n ] are monomials composing f 's numerator and denominator. Then f k = h k g k . Since the corner roots of h and g coincide with the corner roots of h k and g k , respectively, corner integrality is preserved. For the last assertion, we consider two different cases: d(s) = d(t). W.l.o.g., assume d(t) > d(s), thus f (x) d(t)−d(s) = 1 which by Remark 2.3.6, yields that f (x) = 1, so h(x) = g(x). Now, write d = max i∈I {d(i)}, ) d and integrality obviously holds. (2) If x is not as in (1) for exactly one j ∈ {1, ..., m}. If d(j) = 0 corner-integrality at x is trivial, thus we can assume d(j) ≥ 1 for otherwise just take f −1 , and by the above f ′ is corner integral at x as f d(j) is corner-integral (at any point). Since one of the above options is true for any x ∈ R n we get that f ′ is corner-integral.
Remark 9.5.6. It can be shown that if f, g ∈ R(x 1 , ..., x n ) are corner-integral then |f |+|g| may not be corner-integral. Thus the collection of corner-integral principal kernels is not a lattice. In our study we thus take the lattice generated by principal corner-integral kernels which contains elements which are not corner-integral. These elements will be shown to correspond to finitely generated corner loci (to be introduced shortly) which are not principal.
Proof. The assertions follow directly from the construction of Remark 9.6.1 and Corollary 2.7.6. We note that, for α, β ∈ H such that α > β, since α −1 β < 1 we have that (α −1 β) ∔ 1 = 1 and so βf We conclude with the following consequences for R(x 1 , ..., x n ): Corollary 9.6.3. Every principal kernel in R(x 1 , ..., x n ) is a limit of irregular principal kernels. Every skeleton in R n is a limit (with respect to inclusion) of irregular skeletons.
Corollary 9.6.4. Let f be a principal regular kernel in R(x 1 , ..., x n ). Then there exists an irregular principal kernel g such that g ⊂ f and Proof. The existence of g follows from the above discussion.
10 The corner loci -principal skeletons correspondence In this section we will establish a connection between a geometric object, which is a subset of R n , called 'corner-locus' and a certain kind of principal skeletons. In fact, corner locus and what will be shown to be its corresponding principal kernel are two distinct ways to define the exact same subset of R n .
. For monomials f ∈ F (R[x 1 , ..., x n ]) and g ∈ G[x 1 , ..., x n ] one has that x n ] is called the tangible part of f and the polynomial g ν = g ∔ g ∈ G[x 1 , ..., x n ] for g = s j=1 g j is called the ghost part of f . Thus f can be expressed uniquely in the form: In view of this we can consider an element of F (R[x 1 , ..., x n ]) as a polynomial each of whose component monomials occurs either once, if it is a component monomial of the tangible part, or twice, if it is a component monomial of the ghost part. The supertropical polynomial f (x, y) = y ν ∔ x ∔ 1 ν is considered as x ∔ y ∔ y ∔ 1 ∔ 1 where h(x, y) = x is its tangible part and g(x, y) = y ∔ y ∔ 1 ∔ 1 is its ghost part.
We begin by introducing the well known notion of tropical geometry called 'corner root'. In [4,Section (5.2)] Izhakian and Rowen have generalized the notion of (tangible) corner-root to F (R[x 1 , ..., x n ]) as follows: i.e., if f obtains a ghost value at a. This happens in one of the following cases: 1. There exist two distinct monomials f t and f s of f such that f (a) = f s (a) = f t (a).
2. There exists a ghost monomial f t of f such that f (a) = f t (a).
We write Cor(S) for the corner locus defined by S. In the case where S = {f 1 , ..., f r } is finite, we write A = Cor(f 1 , ..., f n ) to indicate that A is a corner locus defined by the mutual corner roots of f 1 , ..., f r ∈ F (R[x 1 , ..., x n ]), and say that A is a finitely generated corner locus.
A corner locus A ⊆ R n is called principal if there exists a supertropical poly- ..., x n ] (i.e., S contains only tangible polynomials). A corner locus not indicated to be regular is assumed to be generalized.
In view of Definition 10.1.8, we define an operator Cor : We now proceed to study the behavior of the Cor operator. As a trivial consequence of Definition 10.1.8 we have the following Proof. First, equality (10.4) is a set-theoretical direct consequence of the definition of corner loci. In turn this implies that Cor( i∈I S i ) ⊆ Cor(S i ) for each i ∈ I and thus Cor( i∈I S i ) ⊆ i∈I Cor(S i ). Conversely, if x ∈ R n is in i∈I Cor(S i ) then x ∈ Cor(S i ) for every i ∈ I, which means that x is a common corner root of {f : f ∈ S i }. Thus x is a common corner root of {f : f ∈ i∈I S i } which yields that x ∈ Cor( i∈I S i ). For the second equation (inclusion) in (10.5), for each j ∈ I, i∈I S i ⊆ S j . Thus, by (10.4), Cor(S j ) ⊆ Cor( i∈I S i ), and so, i∈I Cor(S i ) ⊆ Cor( i∈I S i ). .., f s , g 1 , ..., g t ), Proof. This is a consequence of Definition 10.1.8 and Remarks 10.1.9 and 10.1.10.
As the first equality is obvious, we only prove A ∪ B = Cor {f i · g j } s,t i=1,j=1 . Indeed, for each 1 ≤ i ≤ s and each 1 ≤ j ≤ t, A ⊆ Cor(f i ) and B ⊆ Cor(g j ).
On the other hand, if a ∈ A ∪ B then there exist some i 0 and j 0 such that a ∈ Cor(f i 0 ) and a ∈ Cor(g j 0 ). Thus a ∈ Cor( , proving the opposite inclusion. Remark 10.1.12. In view of Lemma 10.1.11, it is obvious that if A = Cor(f 1 , ..., f s ) and B = Cor(g 1 , ..., g t ) are regular then A ∩ B and A ∪ B are regular (i.e., defined by tangible polynomials).
Definition 10.1.13. Denote the collection of corner loci in R n by CL(R n ) and by RCL(R n ) the family of regular corner loci. Denote the collection of finitely generated corner loci by F CL(R n ) ⊂ CL(R n ) and the collection of principal corner loci by P CL(R n ) ⊆ F CL(R n ). Analogously we denote the collection of finitely generated regular corner loci by F RCL(R n ) ⊂ RCL(R n ) and the collection of principal regular corner loci by P RCL (R n  Taking f = 1 + 1 = 1 ν and g = α with α = 1, we get that R n = Cor(f ) and ∅ = Cor(g) are in F CL (R n

Corner loci and principal skeletons
Having defined a corner locus, we proceed to construct the connection between corner loci and principal skeletons. We begin by constructing it for the special case of principal corner locus. Proof ., x n ]) be a polynomial represented as the sum of its component monomials f i , i = 1, ..., k, where some of them may not be distinct (in fact, appear exactly twice, see Note 10.1.4). Define the following element of R(x 1 , ..., x n ): , all these summands are inessential at x, yielding the first direction of our claim.
Conversely, let f (x) = [m+1] 1 for some x ∈ R, then there are m + 1 summands g i = f i j =i f j , say g 1 , ..., g m+1 such that g 1 (x) = ... = g m+1 (x) = 1. Then, for each i = 1, ..., m + 1, we have f i (x) = j =i f j (x) so there exists at least one essential f j with j = i such that f j (x) = f i (x). Notice that this last observation yields and that also g j (x) = 1. Consequently, there are exactly m + 1 essential monomials at x, f 1 , ..., f m+1 obtaining the same value at x, which in turn yields that x is a corner root of multiplicity m = (m + 1) − 1, as desired. Finally, by the above, Skel( f ) = Cor(f ), i.e., the skeleton defined by f is exactly the corner locus of f .
Proof. If we take f ∈ F (R[x 1 , ..., x n ]) to be non-tangible, say f k and f s are the same monomial g for some 1 ≤ s, k ≤ m, then the terms f k j =k f j and fs j =s f j of f would be irregular since g occurs as the monomial in the numerator and as one of the summands of the denominator. The essentiality of g in f implies that at least one of those terms is essential in f , making it an irregular element of R(x 1 , ..., x n ). Reversing the last arguments yields that if f is irregular then f is non-tangible, i.e., has a component monomial which is ghost.
Proof. If x ∈ R n be such that h i (x) = 1, then there exists k = i such that (10.8) induced by the map f → f given in Proposition 10 Thus, Φ ⋆ extends to a map where Φ : CL(f ) → Skl( f ). In particular, taking only finite generated corner loci, and recalling that finite intersections and unions of principal skeletons are principal skeletons, Φ restricts to the map Φ| F CL(R n ) : F CL(R n ) → P Skl(R n ). As our interest is in the latter map we will denote Φ| F CL(R n ) by Φ.
Note that since the map f → f sends tangible elements of F (R[x 1 , ..., x n ]) (i.e., elements of R[x 1 , ..., x n ]) to regular elements of R(x 1 , ..., x n ) we also have that Φ| F RCL (R n

1, by
Proof. Assume A i = A s , i.e., f i j =i f j = fs l =s f l . Since we have multiplicative cancellation we can multiply both sides by the product of the denominators without changing the equality. Thus we can rephrase this equation as Now, we can partition R n into the following distinct regions: 1. f i is the only essential monomial, in which case (10.10) has the form f 2 i = f s · f i and thus f i = f s which cannot hold (since f i is the only essential monomial).
2. f s is the only essential monomial, in which case (10.10) has the form f i ·f s = f 2 s and thus f i = f s which cannot hold.
3. f i = f s are both essential.
4. Both f i and f s are non-essential, in which case (10.10) has the form f i · A = f s · A and thus f i = f s .
Thus, the only possible solutions are f i = f s both essential and f i = f j < A.
When f i = f s are essential we have that Since there is always some 1 ≤ j ≤ k such that A j ≥ 1 (see the proof of Proposition 10.2.1) we have that both A i and A s are inessential in f . The converse direction of (10.10) is obvious.
Corollary 10.2.6. By Proposition 10.2.3 we can strengthen (10.10) as follows: and are essential in f or A i = A s are inessential in f . (10.11) We now establish the connection between tropical essentiality as described in Definition 9.1.1 and the essentiality introduced in Definition 9.1.5. We show that the map Φ of Remark 10.2.4 sending a finitely generated corner locus to its corresponding principal skeleton, sends tropical inessential terms of a defining polynomial f of a principal locus to inessential terms in the defining element f of the skeleton, and can be generally be omitted from the definition of f without changing the skeleton defined by f .
., x n ]) be a polynomial represented as the sum of its component monomials f for every x ∈ R n and we can write So g m does not surpass any 'roots' of f ′ . Now we have to show that g m does not contribute 'roots' either. Indeed, if g m (x) = 1 then there exists some f s in f with s = m (and so, f l (x) = f s (x)), for otherwise f s and f m are the only monomials defining a corner root of f at x which yields that f m is essential in f , contradicting our assumption.
fs(x) = 1. Note that we could also take g ′ s instead of g ′ l . We have proved that g m and f m could be omitted from f without affecting its skeleton and thus inessential by definition.
Without loss of generality, consider the term This last observation yields that also g s (x) = 1. So, we have shown that Skel(g 1 ) ⊆ Skel( k i=2 g i ). Now, it remains to show that g 1 does not surpass any point of the skeleton defined by k i=2 g i . Assume g j (x) = 1 for some j = 1. Then as above there exists s = j such that f s (x) = f j (x) and f s (x) is dominant in the denominator and f i (x) ≤ f j (x) = f s (x) for all i = j, s. If s = 1 then we get that g 1 (x) ≤ 1 and thus g j is not surpassed by g 1 at x . Thus assume s = 1. In such a case we get g 1 (x) = f 1 (x) f j (x) = 1, and once again g j is not surpassed by g 1 at x. As the occurrence of g 1 does not add or delete any point of the skeleton of k i=2 g i , it can be omitted. Since ., x n ]) be a supertropical polynomial represented as the sum of its component monomials f i , i = 1, ..., k. In Proposition 10.2.1, we considered the fractional function Moreover, by Proposition 10.2.3, f fulfills the condition (9.17) of Lemma 9.2.35, In view of the above, we can say the following: the skeleton off is Cor(f ). By the correspondence between principal kernels and skeletons we have that f = f , or equivalently,f ∼ K f .
As a consequence of Remark 10.2.9, we get the following corollary: ) such that f contains more than 2 (distinct) essential terms g i = f i j =i f j , is a reducible skeleton (with respect to principal skeletons), i.e., the kernel f is reducible (with respect to the sublattice of principal kernels PCon(R(x 1 , ..., x n ))).
Proof. Let f be in reduced form.
Assume that f is irreducible, then there exists 1 ≤ t ≤ k such that Skel( f ) = Skel(g t ). Then Skel(g t ∔ k i=1;i =t g i ) = Skel( f ) = Skel(g t ), and thus k i=1;i =t g i is inessential, contradicting our assumption that f contains more than two essential terms g i .
Thus f is irreducible and so corresponds to an irreducible kernel. In the case where f = f 1 is a single monomial, f translates to f = f 1 0 ∔ 0 f 1 = |0|, and thus { f = 1} = ∅.
We now give an example of applying our theory to the well-known tropical line This form is not reduced, since each of the component fractions can be omitted from the sum without affecting the skeleton of f . Geometrically these three terms correspond to the three angles formed by the tropical line, of which one can be evidently omitted. f is corner integral and regular and thusf = y x∔1 ∧ x y∔1 ∧ 1 x∔y is also a generator of f , i.e, f ∼f . In reduced form, we can take f = y x∔1 ∔ x y∔1 or f = y x∔1 ∔ 1 x∔y or f = 1 x∔y ∔ x y∔1 . Say we take the former. Thenf = y x∔1 ∧ x y∔1 . It can be easily seen that no more than one of the three terms introduced above can be omitted from f without changing the skeleton.
., x n ) having monomial numerator for i = 1, ..., k. We say that f admits the bound property if for any x ∈ R n the following condition holds: The latter option means that f i and f j are not essential in f at x.
Proof. Clearly, the second assertion follows the first, by taking inverse elements.
., x n ] are monomials composing h's numerator and denominator. Then h = k i=1 ( f i g ). Assume there exists some . So by the boundary property this yields that either f i (x) g(x) are inessential in h. The former yields that f i (x) = g(x) = f j (x) so there is some t ∈ {1, ..., m} such that f i (x) ≤ g t (x). The latter implies that f i (x) is inessential, as desired. Proof. As a consequence of Lemma 10.2.5, we get that f admits the bound property. Thus by Remark 10.2.14 we have that condition (11.1) holds. For condition (11.2) . We want to show that every corner root of g is surpassed by some summand of h. First note that the corner roots of g are exactly those of the polynomial w = k i=1 f i . Indeed, if x ∈ R n is a corner root of w then is a corner root of precisely k − 2 factors ( j =i f j ) with i an index of any chosen pair of essential monomials. Conversely, if x ∈ R n is a corner root of g then it is a corner root of k − 2 of its factors not involving the essential monomials, say f i and f j , which yields essentiality of f i and f j in w, proving our claim. In view of the last assertion, we can restrict our attention to the corner roots of w. Now, assume f i (x) = f j (x) for some i, j ∈ {1, ..., k}. If there exists s ∈ {1, ..., k} \ {i, j} such that f s (x) > f i (x), take the one attaining the maximal value at x, the f s ( q =s ( l =q f q (x))) ≥ g(x). If no such s exists, then By the construction of f and Corollary 10.2.2, irregularity of f implies a multiple occurrence of the same monomial in f , so f is non-tangible thus the second assertion follows.
. (10.12) Let Z = Skel(h) ⊆ R n be a principal corner integral skeleton. Then Z corresponds to a corner locus of the supertropical polynomial h ∈ F (R[x 1 , ..., x n ]), in the sense that the generalized corner locus Cor(h) coincides with Z in R n . If h is also regular then h is tangible so Cor(h) ∈ P RCL(R n ) is a regular corner locus.
Proof. The corner locus of (10.12) is As h is corner integral we have that as all corner roots of the numerator and denominator of h are surpassed. Now, the skeleton defined by h is If h is taken to be also regular, then all multiple occurrences of monomials in h = k i=1 f i ∔ m j=1 g j are inessential and can be omitted, and thus h is tangible.
Definition 10.2.17. Let f, g ∈ F (R[x 1 , ..., x n ]) be a pair of supertropical polynomials. Define the following relation The relation ∼ L is obviously reflexive, symmetric and transitive, thus is an equivalence relation on F (R[x 1 , ..., x n ]).
Proposition 10.2.18. There is a 1 : 1 correspondence between principal cornerintegral skeletons and principal corner-loci which restricts to a correspondence between principal regular corner-integral skeletons and principal regular corner-loci. This correspondence induces a correspondence between principal (regular) corner-integral kernels and principal (regular) corner-loci.
Proof. Let f ∈ R(x 1 , ..., x n ) be corner integral, and let f ′ ∼ K f . Then Conversely, Let g ∈ F (R[x 1 , ..., x n ]) and let g ′ ∼ L g. Then where the last equality holds since we have shown g to be corner integral. So The restriction to regular skeletons, their corresponding kernels and corner loci follows the propositions introduced above.
Definition 10.2.19. Let Ω be the lattice generated by principal corner integral kernels with respect to (finite) multiplications and intersections.
Remark 10.2.20. Now, using the procedure introduced in Remark 10.2.4 and the correspondence introduced in Corollary 7.0.26, we get that Ω corresponds to the lattice of finitely generated generalized corner loci.
Intersecting Ω with the lattice of regular kernels yields a lattice Θ ⊂ PCon(R(x 1 , ..., x n )) generated by regular corner-integral principal kernels which corresponds to the lattice of regular finitely generated corner loci F RCL(R n ).
Note 10.2.21. We are mainly interested in the sublattice Θ ⊂ PCon( R ) of corner-integral principal kernels in Con( R ), for which the above holds too. Θ also is a sublattice of the lattice of corner-integral principal kernels in PCon(R(x 1 , ..., x n )).
Remark 10.2.22. Θ defined above is a sublattice of kernels of PCon(R(x 1 , ..., x n )) (or of PCon( R )). Thus all results of the section concerning reducibility is applicable to it.
In the subsequent section concerning kernel dimension, we will show that the lattice generated by corner integral regular kernels is in fact the lattice of regular kernels! Thus the lattice of corner loci corresponds to the lattice of regular kernels.

Example: The tropical line
Note 10.3.1. In the following example we consider the rational function f = x y∔1 ∔ y x∔1 ∔ 1 x∔y ∧|α| ∈ R for any α ∈ R\{1}. As taking ∧|α| does not affect the computations below, we prefer omitting it, and work with x y∔1 ∔ y x∔1 ∔ 1 x∔y instead of its 'copy' in R .
Example 10.3.2. Let f = x∔y∔1 be the tropical line. Its corresponding skeleton is defined by the rational function f = x y∔1 ∔ y x∔1 ∔ 1 x∔y , and so, its corresponding kernel in R(x, y) is f . As shown above f ∼ K x y∔1 ∧ y x∔1 ∧ 1 x∔y . Moreover, any of the three terms above can be omitted. Thus we have that where each of the kernels comprising the intersection is contained in both of the remaining kernels (in the last kernel we chose to take x ∔ y as a generator instead of its inverse). Now, taking logarithms, it can be seen that Skel( x∔y ) is exactly the union of the bounding rays of the third quadrant. As we have that x ∔ y = |x ∔ y| ∔ |x| ∩ |x ∔ y| ∔ |y| . Notice that 1 x∔y does not admit the corner integrality condition, as the corner roots of x and y are not surpassed by 1, thus does not correspond to a tropical hypersurface. We proceed by considering x y∔1 ∔ y x∔1 omitting the last term of f . Computing its representation as a single fraction, we get x 2 ∔x∔y 2 ∔y (x∔1)(y∔1) = (x∔y)(x∔y∔1) (x∔1)(y∔1) . The corner roots of the numerator are {x = 1, y = 1} and {x = y}, and the corner roots of the denominator are {x = 1} and {y = 1}. The corner root {x = y} is surpassed by the denominator as (x∔1)(y∔1) = (x∔y)∔(xy∔1) ≥ (x∔y) thus this is a regular skeleton (or equivalently a regular kernel) as expected. We urge the reader not to try to put negative values into the last equation, since they, of course, do not exist in a semifield. Finally, we discuss the above wedge decomposition of |x∔y|. It is a quite natural one. The geometric locus of the equation |x|∧|y| = min(|x|, |y|) = 1 in a logarithmic scale is exactly the axes, as a union of the x-axis corresponding to |x| = 1 and the y-axis corresponding to |y| = 1. Intersecting it with the geometric locus of |x ∔ y| leaves the latter untouched as the former locus contains it. In fact, using such methods of intersections we can define any segment and ray in R 2 using principal kernels, so only points in the plane are irreducible skeletons. This of course is not a problem, since we are still free to consider lattices inside the lattice of principal kernels, generated by designated subsets, which will be considered prime or irreducible. In fact, principal kernels leave us with maximal 'flexibility' in our hands.

Corner-integrality revisited
In this section we study the notion of corner-integrality introduced in sub section 9.5. We specify a procedure for finding a corner-integral kernel containing a given kernel or equivalently a supertropical hypersurface containing a given skeleton. We show that under this procedure corner-integral kernels are left unchanged.
Let f ∈ R(x 1 , ..., x n ) be a rational function. We start by expressing cornerintegrality of f and thus of f , the kernel generated by f , by a condition involving kernels. Write f = h g , where h = k i=1 h i and g = m j=1 g j where h i and g j for i = 1, ..., k and j = 1, ..., m are the component monomials in R[x 1 , ..., x n ] of the numerator and denominator of f , respectively. For simplicity, we assume that f is in essential form, so each monomial affects Skel(f ), the skeleton defined by f . As noted in Subsection 9.5, f is corner-integral if the following two conditions hold for every x ∈ R n : Equivalently, f is corner integral if condition (11.1) holds for both f and f −1 (exchanging the g i 's and the h j 's in the condition).
Proposition 11.0.3. Let f ∈ R(x 1 , ..., x n ). Then f is corner integral if and only if |f | is corner-integral.
Proof. Write f = h g where f, g ∈ R[x 1 , ..., x n ]. Then |f | = h g + g h = h 2 +g 2 gh . Since |f | ≥ 1 we only need to check that the corner roots of the numerator of |f | are surpassed by its denominator. If f is corner integral then in the numerator all corner roots of g are surpassed by h and vise versa so we are left with the scenario of a corner root a such that h 2 (a) = g 2 (a) which yields that h(a) = g(a) (note that it is possible that a will also be a corner root of g 2 or h 2 ). In such a case h 2 (a) = g 2 (a) = gh(a) thus the value of the numerator of |f | at a is surpassed by the value of the denominator, proving that |f | is corner-integral. Conversely, assume |f | is corner integral then h 2 + g 2 is surpassed by gh, in particular any corner root of h 2 is surpassed by gh which yields that any corner root of h is surpassed by g, and any corner root of g 2 is surpassed by gh which yields that any corner root of g is surpassed by h, thus f is corner-integral.
Proof. Since f and g are corner integral we have that |f | and |g| is corner-integral.
Since f and g are corner-integral the corner roots of f 1 are surpassed by f 2 and the corner roots of g 1 are surpassed by g 2 . Thus the corner roots of f 1 g 1 are surpassed by either f 2 g 1 or f 1 g 2 . We note that there may be equalities of the form f 1,u for some x ∈ R n , but these 'corner roots' are surpassed by are dominant monomials of f 1 and g 1 at x. In the opposite direction, since |f | ≥ 1 and |g| ≥ 1 we have that f 1 ≥ f 2 and g 1 ≥ g 2 , respectively. Thus f 1 g 1 ≥ f 1 g 2 , f 2 g 1 so f 1 g 1 ≥ f 1 g 2 + f 2 g 1 and so in particular f 1 g 1 surpasses all corner roots of f 1 g 2 + f 2 g 1 .
Corollary 11.0.5. An intersection of corner-integral kernels is a corner-integral kernel.
Proof. Let K 1 and K 2 be corner-integral (principal) kernels and let u 1 and u 2 be a pair of corner-integral generators for K 1 and K 2 , respectively. By Proposition 11.0.3 we may assume u 1 , u 2 ≥ 1. Take f = u 1 ∧ u 2 , then by Proposition 11.0.4 f = u ∧v = |u| ∧|v| is corner-integral thus by definition f is corner-integral.
Proof. By Proposition 11.0.3 we may assume f ≥ 1, i.e. f = |f |. Since |α| is trivially corner-integral by Proposition 11.0.4 the corner-integrality of f implies the corner-integrality of f ∧ |α|. Conversely, if f is not corner-integral then Recall that a k-kernel is a kernel in Con(R(x 1 , ..., x n )) which is a preimage of a skeleton Z = Skel(S) ⊆ R for some subset S ⊂ R(x 1 , ..., x n ) under the map Ker : P(R n ) → R(x 1 , ..., x n ). We have shown that Skel(S) = Skel( S ) where S is the kernel generated by the elements of S. We have found these k-kernels for the restriction Ker| R : P(R n ) → R . We have shown that the preimage in R of a principal skeleton to be Ker(Skel(f )) = f ∩ R so that for α = 1 Ker(Skel( |f | ∧ |α| )) = |f | ∧ |α| = f ∩ R . Since corner integrality of f implies corner integrality of |f | ∧ |α| we refer to f rather then to f ∩ R in our computations.
We will now translate condition (11.1) introduced above to the language of kernels. Let C i (h) be the set of corner roots of h attained by the monomial h i (and some other monomials) and denote by h i = j =i h j . Then condition (11.1) is equivalent to saying that if x ∈ R n is a corner root of h then h(x) ≤ g(x) (i.e g surpasses h at x). So, for every 1 ≤ i ≤ k we can formulate the condition by This means that f is corner-integral if all the corner roots of its numerator h are contained in the skeleton defined by f + 1, i.e. in the region of R over which f ≤ 1. Since Skel(K 1 ) ⊆ Skel(K 2 ) ⇔ K 2 ⊂ K 1 for any pair of K-kernels K 1 , K 2 we have 11.4) Now, intersecting both sides of the expression obtained in (11.3) by Note that the second transition is an equivalence rather then implication since Again, translating the resulting skeletons expression to kernels yields Moreover, since the above inclusion holds for every i ∈ {1, ..., k} we have that and so Interchanging the roles of h and g we get that the second condition 11.2 translates to which is equivalent to (11.5) and so since the inclusion in (11.5) holds for every j ∈ {1, ..., m} we have that m j=1 Skel g j g j ∩ Skel(f + 1) ⊆ Skel(f ) Using the notation introduced above we have the following necessary and sufficient conditions hold: and similarly m j=1 Skel g j g j = Skel(g).
Thus we can rewrite (11.6) and (11.7) in the form or in the form Finally, we have proved that f is corner-integral for any supertropical polynomial f ∈ F (R[x 1 , ..., x n ]).
In view of Proposition 11.0.7, given f = h g ∈ R(x 1 , ..., x n ), in order to obtain a corner-integral fraction whose skeleton contains Skel(f ) one must adjoin both to the skeleton of f .
Define the map Φ CI : R(x 1 , ..., x n ) → R(x 1 , ..., x n ) by taking Φ CI (f ), where f = h g , to be the fraction whose skeleton is formed by adjoining all the necessary points required for f to admit corner integrality to Skel(f ). Namely Φ CI (f ) = |f | ∧ |f −1 + 1| +h ∧ (|f + 1| +g) . (11.8) By this definition we have that Proposition 11.0.9. Let f = h g ∈ R(x 1 , ..., x n ) be a rational function, where h = k i=1 h i and g = m j=1 g j where h i and g j for i = 1, ..., k and j = 1, ..., m are the component monomials in R[x 1 , ..., x n ] of the numerator and denominator of f , respectively. Then (11.9) By Remark 9.5.5, if f is corner-integral then so is In particular this applies to f ′ = f −1 + f = |f |. It turns out that this equality Φ CI (f ) = Φ CI (|f |) holds for any f as we prove in the following remark.

Proof. Due to the Frobenious property
Since s + t = 0 we have that h + g s+t is a generator of h + g thus The latter equality holds since Corollary 11.0.13. Let f ∈ R(x 1 , ..., x n ) be such that f = u 1 ∧ · · · ∧ u k where u 1 , ..., u k ∈ R(x 1 , ..., x n ) are each corner-integral. The f is corner-integral and Proof. f is corner-integral by induction on Proposition 11.0.4. First we assume f ≥ 1 (thus so are the u i 's). Since (11.12) holds. For any g ∈ R(x 1 , ..., x n ) taking || does not change corner-integrality and Φ CI (|g|) ∼ K Φ CI (g). Thus for any f we can consider |f | instead and apply the first case.
First note thatf ≥ 1. Now, for any given i ∈ {1, ..., k} denote D i = f i j =i f j , then by Remark 11.0.12 Remark 11.0.15. An R-homomorphic image of corner-integral element of R(x 1 , ..., x n ) is corner-integral. Thus the R-homomorphic image of a cornerintegral kernel is corner-integral.

Composition series of kernels of an idempotent semifield
In this section, we restrict our discussion to idempotent semifields. We write an analogue to the theory of composition series of modules, just for kernels of an idempotent semifield. The kernels of an idempotent semifield are also subsemifields, thereby allowing us to utilize the isomorphism theorems to prove our assertions.
Remark 12.0.16. Let S be an idempotent semifield and let K 1 , K 2 be kernels of S such that K 1 ⊆ K 2 . Then K 1 is a kernel of K 2 . In such a case we say that K 1 is a subkernel of K 2 and write K 1 ≤ K 2 . If K 1 is strictly contained in K 2 , we write Proof. K 2 is a subsemifield of S, so, by Theorem 2.2.51.(1), Remark 12.0.17. As it was previously shown, in Corollary 2.7.7, the family of principal kernels of an idempotent semifield is a sublattice of kernels. Moreover, By remark 2.3.19, homomorphic images of principal kernel are principal kernels. Thus the Isomorphism Theorems 2.2.51 and 2.2.52 hold for principal kernels of an idempotent semifield. Consequently, all subsequent assertions hold for principal kernels too (considering idempotent semifields).
Definition 12.0.18. Let L be a kernel of an idempotent semifield S. A descending chain The factors of the chain are the kernels K i−1 /K i , for 1 ≤ i ≤ t. The chain in (12.1) is said to be a composition series for L if K t = 1 and each factor is a simple kernel.
We say that two chains of kernels are equivalent if they have isomorphic factors.
Remark 12.0.19. Let L be a kernel of an idempotent semifield S.
If L = K 0 ⊃ K 1 ⊃ · · · ⊃ K t is a chain C of kernels and K ≤ K t , then the chain C ′ given by L/K = K 0 /K ⊃ K 1 /K ⊃ · · · ⊃ K t /K is equivalent to C.
Proof. By Theorem 2.2.52 we have that K i−1 /K i ∼ = (K i−1 /K)/(K i /K), and thus the factors of C and C ′ are isomorphic.
Remark 12.0.20. If K i−1 ⊃ K i are kernels such that the semifield K i−1 /K i is not simple, then there exists some kernel N between them, i.e., K i−1 ⊃ N ⊃ K i . The process of inserting such an extra subkernel N into the chain is called refining the chain.
Consequently, any chain that is not a composition series can be refined.
Remark 12.0.21. For any simple subkernel S and any K < L, by Theorem 2.2.51. (2), we have that which is either isomorphic to S or 1 = {1}. It follows that L is a finite product of simple subkernels Remark 12.0.22. Let L be a kernel of an idempotent semifield S. Define a composition chain C(L, K) from L to a subkernel K to be a chain such that each factor is simple. By Remark 12.0.19, C(L, K) is equivalent to the composition series L/K ⊃ L 1 /K ⊃ · · · ⊃ L t /K = 1 of L/K. It follows that if P is a subkernel of a kernel N for which N/P ∼ = L/K, then there is a composition chain C(N, P ) equivalent to C(L, K).
The following is the well-known Jordan Hölder theorem for kernels: Theorem 12.0.23. Let L be a kernel of an idempotent semifield S. Suppose L has a composition series which we denote by C. Then 1. Any arbitrary finite chain of subkernels L = K 0 ⊃ K 1 ⊃ · · · ⊃ K s (denoted as D), can be refined to a composition series equivalent to C. In particular s ≤ t.
2. Any two composition series of L are equivalent.
3. ℓ(L) = ℓ(K) + ℓ(L/K) for every subkernel K of L. In particular, every subkernel and every homomorphic image of a kernel with composition series has a composition series.
Proof. We prove the theorem by induction on t. If t = 1 then L is simple and the theorem is trivial, so we assume the whole theorem is true for kernels having a composition series of length ≤ t − 1.
(1) Let C 1 = C 1 (L 1 ) denote the composition series L 1 ⊃ · · · ⊃ L t = 1 of L 1 ; C 1 has length t−1. If K 1 ⊆ L 1 , then by induction on t, the chain L 1 ⊇ K 1 ⊃ K 2 ⊃ · · · ⊃ K s can be refined to a composition series of L 1 equivalent to C 1 , yielding (1) at once (by tagging on L ⊃ L 1 ). Thus, we may assume Also, by Remark 2.5.7, L 1 · K 1 = L since L 1 is maximal in L. Note that we have two ways of descending from L to L 1 ∩ K 1 ; either via L 1 or via K 1 . But these two routes are equivalent, in the sense that By induction on t, the chain L 1 ⊃ L 1 ∩ K 1 ⊃ 1 refines the composition series E 1 (L 1 ) equivalent to C 1 (of length t − 1). This is comprised of E ′ 1 (L 1 , L 1 ∩ K 1 ), a composition series from L 1 to L 1 ∩ K 1 of some length t 1 followed by a composition series E ′ 2 (L ∩ K 1 ) of some length t 2 , where t 1 + t 2 = t − 1. Since t 1 ≥ 1, we see t 2 ≤ t − 2. Furthermore, since L 1 is maximal in L, by Corollary 2.5.8 we have that L/L 1 is simple, thus (12.3) shows that K 1 /(L 1 ∩ K 1 ) is simple. So the chain K 1 ⊃ L 1 ∩ K 1 followed by E ′ 2 is a composition series F 1 of K 1 having length t 2 + 1 ≤ t − 1. By induction, the chain K 1 ⊃ K 2 ⊃ · · · ⊃ K s refines to a composition series D 1 (K 1 ) equivalent to F 1 . The isomorphism (12.2) enables us to transfer E ′ 1 (L 1 , L 1 ∩ K 1 ) to an equivalent composition series E ′′ 1 (L, K 1 ) from L to K 1 , also of length t 1 . Tracking this onto D 1 (K 1 ) yields the desired composition series refining D which is equivalent to C. In conclusion, we have passed from our original composition series C 1 through the following equivalent composition series: 1. L ⊃ L 1 followed by E ′ 1 (L 1 , L 1 ∩ K 1 ) and E ′ 2 ; 2. E ′′ 1 (L, K 1 ) followed by K 1 ⊃ L 1 ∩ K 1 and E ′ 2 ; 3. E ′′ 1 (L, K 1 ) followed by D 1 , which defines D, as desired.
(3) Refine the chain L > K > 1 to a composition series, and apply Remark 12.0.22.
Definition 12.0.24. Given a kernel L of an idempotent semifield S, we define its composition length ℓ(L) to be the length of a composition series for L, if such exists.
Remark 12.0.25. Let S be an idempotent semifield. All the results stated in this section hold taking any sublattice of kernels Θ of Con(S) in the sense of Definition 9.2.1, instead of Con(S). Note that, given Θ, maximal kernels are taken to be maximal Θ-kernels, i.e., maximal elements of Θ. For example, one can consider the sublattice of principal kernels of S, PCon(S).
13 The Hyperspace-Region decomposition and the Hyperdimension 13.1 Hyperspace-kernels and region-kernels.
Remark 13.1.1. Though we consider the semifield of fractions R(x 1 , ..., x n ), most of the results introduced in this section are applicable to any finitely generated semifield R(a 1 , ..., a n ) over R, where {a 1 , ..., a n } are generators of R(a 1 , ..., a n ) as a semifield. We explicitly indicate whenever a condition needs to be imposed on {a 1 , ..., a n } to hold for the semifield R(a 1 , ..., a n ). In particular, R ⊂ R(x 1 , ..., x n ) is just another case of a finitely generated semifield over R, taking a i = x i ∧ |α| for 1 ≤ i ≤ n and α ∈ R \ {1}. In this case, the generators a i are bounded from above (or simply |a i | are bounded), and we specifically designate the results that are true only for unbounded generators. g . This is consistent with our interest in proper subkernels in R . We could equivalently take f to be an element of R .
Remark 13.1.4. One can choose to view an HP-fraction simply as a nonconstant Laurent monomial in R(x 1 , ..., x n ).
Remark 13.1.5. HP-fractions in R(a 1 , ..., a n ) where a i are unbounded fractions, are not bounded; i.e., for any HP-fraction f there exists no α ∈ R such that |f | ≤ |α|.
Proof. Let f ′ = h g where h, g ∈ R[x 1 , ..., x n ] are distinct monomials such that h g ∩ R ⊆ {1}. Then f ′ contains at least one free (unbounded) indeterminate. Thus f ′ = 1 and so |f ′ | ≤ 1. So we may assume α = 1. Now, since R is divisibly closed there exists some a k ∈ R n such that f ′ (a k ) = |α| k for any natural k.
Proof. The claim follows from the inequality |f j | ≤ t i=1 |f i | = f ′ and the invariance of boundness under ∼ K .
Definition 13.1.11. A principal kernel of R(x 1 , ..., x n ) is said to be a hyperplanefraction kernel (or shortly, HP-kernel ) if it is generated by a hyperplane fraction.
Remark 13.1.12. An HP-Kernel is regular.
Indeed, as regularity is preserved under ∼ K we may assume f = h g with h and g distinct monomials as both numerator and denominator of a hyperplane fraction are monomials, the condition for regularity holds trivially.
Definition 13.1.13. A principal kernel of R(x 1 , ..., x n ) is said to be a hyperspacefraction kernel (or shortly, HS-kernel ) if it is generated by a hyperspace fraction.
Remark 13.1.14. A principal kernel is an HS-kernel if and only if it is a product of HP-kernels.
Proof. Every hyperplane fraction is a hyperspace fraction comprised of a single summand, thus every HP-kernel is an HS-kernel. Conversely, if f is an HS- ..., x n ] are distinct monomials for each 1 ≤ i ≤ t. Thus, by definition f i is an HP-kernel for each i ∈ {1, ..., t} proving our claim.
Corollary 13.1.15. An HS-Kernel is regular.
Proof. Since an HP-kernel is regular and since, by Remark 13.1.14, an HS-kernel is a product of HP-kernels, the assertion follows by Remark 9.4.4 and Corollary 9.4.9.
Definition 13.1.16. A skeleton in R n is said to be a hyperplane-fraction skeleton (shortly HP-skeleton) if it is defined by a hyperplane fraction. A skeleton in R n is said to be a hyperspace-fraction skeleton (shortly HS-skeleton) if it is defined by a hyperspace fraction.
Corollary 13.1.17. A skeleton is an HS-skeleton if and only if it is an intersection of HP-skeletons.
Proposition 13.1.18. Let f be an HP-kernel. Then w ∈ f is an HP-fraction if and only if w s = f k for some s, k ∈ Z \ {0}. Assume w s = f k for any s, k ∈ Z \ {0}. We will show that there exists some a ∈ R n such that a ∈ Skel(f ) \ Skel(g) for some g ∈ f . Let (p 1 , ..., p n ), (q 1 , ..., q n ) ∈ Z n be the vectors of powers of x 1 , ..., x n in the Laurent monomials f and w. Since w and f are nonconstant, (p 1 , ..., p n ) = (0),(q 1 , ..., q n ) = (0). By Remark 2.3.7 we may assume that gcd(p 1 , ..., p n ) = gcd(q 1 , ..., q n ) = 1 (since R is divisible, the constant terms of f and w can be adequately adjusted). Since w ∈ f we have that |w| ≤ |f | m for some m ∈ N, so if x i occurs in w it must also occur in f . Finally, since w s = f k for any k ∈ Z\{0} we can also assume that (p 1 , ..., p n ) = (q 1 , ..., q n ), for otherwise w = αf for some α = 1 and thus contradicting our assumption that Skel(w) ⊇ Skel(f ) and Skel(w) = ∅. Let x j l occur in w for some l ∈ Z \ {0} such that x j is not identically 1 over Skel(w) (and thus also on Skel(f )). Thus there exists some k ∈ Z \ {0} such that x j l+k occurs in f . Define the Laurent monomial g = w −1 f k ∈ f , then x j does not occur in g. Without loss of generality, assume j = 1 where the exponent of x 1 in f is p 1 .
If a = (α 1 , ..., α n ) ∈ Skel(f ), then g(a) = w(a) −1 f (a) = 1. By our assumption that w s = f k there exists x t occurring in f with some power p t ∈ Z \ {0} and not in w (for otherwise they both contain only x j ). Take b = (1, α 2 , ..., β, ..., α n ) with β ∈ R occurring in the t'th place, such that β pt = α p t t α p 1 1 (there exists such β since R is divisible). Then as x j is not identically 1 over Skel(f ), we can choose a ∈ Skel(w) such that f (b) = 1 and g(b) = 1.
Corollary 13.1.19. Let f be an HP-kernel. Then w ∈ f is an HP-fraction if and only if w is a generator of f .
Proof. The claim follows from Remark 13.1.18 and the property that g k = g for any principal kernel of a semifield.
Definition 13.1.20. We define an order-fraction o in the semifield R(x 1 , ..., x n ) to be an element of the form o = 1 ∔ f for some HP-fraction f = 1. We say that o is the order fraction defined by f . Proof. Take O c = 1∔f −1 then since f is an HP-fraction so is f −1 and thus O c is an Remark 13.1.26. We will now clarify the reason behind the last condition posed on the order-fractions comprising the region fraction. For i = 1, ..., t let o i = 1∔g i with g i the HP-fraction defining the order fraction o i . Then since 1 ∔ f i ≥ 1 for every i, we have that Thus a region-fraction r can be defined as r ∼ K 1 ∔ t i=1 f i , so the last condition of the definition can be stated as f j = f −1 i for any 1 ≤ i, j ≤ t. One can see that if there exist k and m for which . We aim for a region fraction to define a skeleton containing some neighborhood in R n , thus the latter condition is required by the above discussion.
Definition 13.1.27. A principal kernel R ∈ PCon(R(x 1 , ..., x n )) is said to be a region kernel if it is generated by a region fraction.
Remark 13.1.28. R is a region kernel if and only if it is of the form Proof. Let r be a generating region fraction of R. So, Since o i is an order fraction for each 1 ≤ i ≤ v, we have by definition that the Lemma 13.1.29. Any HP-kernel is corner-integral, and any order kernel is corner-integral.
Proof. As an HP-fraction has a single monomial in its numerator and denominator, there are no corner roots to surpass, and thus it is trivially corner integral.
For an order kernel, as it is generated by an element of the form 1 ∔ h g = g∔h g for monomials g and h, it has a single corner root at the numerator (x such that g(x) = h(x)) which is surpassed by the denominator g(x). Finally as corner integrality does not depend of the choice of the generator, our claim is proved.
Remark 13.1.30. If f = 1 is an HP-kernel and g is an order kernel, then f · g = |f | ∔ |g| is regular.
Proof. As regularity does not depend on the choice of a generator of the kernel and since |f |∔|g| = f · g , we may consider |f |∔|g|. Write f = u v and g = 1∔ a b with u, v, a and b monomials in for any x ∈ R n . By assumption, f = 1 and thus u = v, so u 2 b = uvb and v 2 b = uvb. Thus for any x ∈ R n there is always a monomial in the numerator distinct from the one in the denominator. Note that auv dominates , and thus essentiality of auv implies auv = buv, i.e., a = b concluding our proof.

Geometric interpretation of HS-kernels and region
kernels and the use of logarithmic scale Definition 13.2.1. Let γ ∈ H, and let k ∈ N. A k-th root of γ, if exists, is an element β ∈ H such that β k = α.
Remark 13.2.2. For any k ∈ N and γ ∈ H the k-th root of γ is unique.
Let (H, ·, ∔) be a divisible semifield. By Remark 13.2.2, we can uniquely define any rational power of the elements of H. In such a way, H becomes a vector space over Q, rewriting the multiplicative operation · on H as addition and defining (m/n) · α = α m n . (13.1) In this way we can apply linear algebra techniques to (H, ·). When considering H in such a way we will denote the original addition of H (in the idempotent case) by ∔ or ∨ in order to avoid ambiguity. We call the representation given by (13.1) the logarithmic representation of (H, ·). H viewed as just described, any HP-fraction may be considered as a linear functional over Q and thus an HP-kernel may be considered as a kernel generated by a linear functional (notice that for an HP-fraction f , the equation f = 1 over (H, ·) n translates to F = 0 over (H, +) n where F is the linear form obtained from f by applying (13.1), i.e., F is the logarithmic form of f .) Remark 13.2.3. In the special case of a semifield R, by Theorem 3.4.13, the above interpretation allows us to consider (R, ·) as being (R, +) and R n as being R n with coordinate-wise addition and scalar multiplication over Q.
Proof. Writing f (x 1 , ..., x n ) = α n i=1 x k i i with k i ∈ Z, then α = f (a 0 ) n i=1 a −k i 0,i . After α is determined, since p 0 , ...., p n are in general position the set {a 1 , ..., a n , b = (f (a 1 ), ..., f (a n ))} ⊂ R n define a linearly independent set of n linear equations in the variables k i , and thus determine them uniquely.
Consider an HS-kernel of R(x 1 , ..., x n ) defined by the HS-fraction f = t i=1 |f i | where f 1 , ..., f t are HP-fractions. Then f = 1 if and only if f i = 1 for each i = 1, ..., t. Thus f = 1 gives rise to a homogenous system of rational linear equations of the form F i = 0 where F i is the logarithmic form of f i . This way Skel(f ) ⊂ R n ∼ = (R + ) n is identified with an affine subspace of R n which is just the intersection of the t affine hyperplanes defined by F i = 0, 1 ≤ i ≤ t. Analogously, for an order kernel defined by o = 1 ∔ g for some HP-fraction g, o = 1 if and only if g ≤ 1 which gives rise to the rational half space of R defined by the weak inequality g ≤ 0. Thus, the region kernel defined by

A preliminary discussion
We introduce the following example to motivate our subsequent discussion. Consider a point a = (α 1 , ..., α n ) ∈ R n . Then a = Skel(f a ) where We would like f a to encapsulate the dimension reduction from R n to {a}. Consider the following chain of principal HS-kernels For each 1 ≤ k ≤ n denote f k = x 1 α 1 ∔ · · · ∔ x k α k and f 0 = 1. Each of the HS-kernels f k−1 in the chain is also a semifield which is a subsemifield of the preceding kernel f k−1 . The factors defined by the quotients f k / f k−1 are the quotient semifields Note that on the right hand side of the equality we have a homomorphic nontrivial image of an HP-kernel, namely, an HP-kernel of the quotient semifield There are some questions arising from the above construction: Can this chain of HS-kernels be refined to a longer descending chain of principal kernels descending from f a ? Are the lengths of descending chains of principal kernels beginning at f a bounded, and if so, can any chain can be refined to a chain of maximal length? The following example provides a positive answer to the first question: Consider the kernels |x| ∔ |y| and |x| = x . Both are semifields over the trivial semifield and |x| is a subkernel of |x| ∔ |y| . Consider the substitution map φ sending x to 1. Then Im (φ) = 1 ∔ |y| = |y| R(y) . The kernel |y| is not simple in the lattice of principal kernels of the semifield |y| R(y) as we have the chain |y| ⊃ |1 ∔ y| ⊃ 1 , which is the image of the refinement . One can notice that 1 ∔ y is an order kernel which induces the order y ≤ 1 on the semifield |y| .
In view of the above example we would like to restate the questions posed above as follows: Can this chain of HS-kernels be refined to a longer descending chain of HS-kernels descending from f a ? Are the lengths of descending chains of HS-kernels beginning at f a bounded, and if so, can any chain of HS-kernels be refined to such a chain of maximal length?
In the next section we provide answers to these three questions by which the chain introduced above is of maximal unique length common to all chains of HS-kernels descending from f a .

The HO-decomposition
In the following we describe an explicit decomposition of a principal kernel f as an intersection of kernels of two types: The first, to be named an HO-kernel, is a product of some HS-kernel and a region kernel. The second is a product of a region kernel and a bounded from below kernel.
While the first type defines the skeleton of f , the second type has no effect on it as it corresponds to the empty set. This latter type is the source of ambiguity in relating a skeleton to a kernel, preventing the kernel corresponding to a skeleton from being principal. When intersecting with R , the kernels of the second type in the decomposition are chopped off, in the sense that they all become equal to R . This restriction to R thus removes the ambiguity making each HO-kernel (intersected with R ) in a 1 : 1 correspondence with its skeleton (which is, in fact, a segment in the skeleton defined by f ). Subsequently, the 'HO-part' is unique and independent of the choice of the kernel generating the skeleton.
Geometrically, the decomposition to be described below is just a fragmentation of a principal skeleton defined by f to the "linear" fragments comprising it. Each fragment is attained by bounding an affine subspace of R n defined by an appropriate HS-fraction (which in turn generates an HS-kernel) using a region fraction (generating a region kernel). Note that the HS-fraction may be 1 and so may the region fraction.
By the discussion above, although the HS-fraction and region fraction defining each segment may vary moving from one generator of the principal kernel to the other, the HS-kernels and region kernels they define stay intact as they correspond to the fragments of the skeleton of f . Thus we may form the next construction, though explicit, using any generator without affecting the resulting HO-kernels.
We now move forward to introduce the construction which will be illustrated by two subsequent examples.
Consider an element f ∈ R(x 1 , ..., x n ), the kernel it generates, f , and its corresponding skeleton Skel(f ). Taking |f | we may assume f ≥ 1. Write and a subset G a ⊆ G = {g j : 1 ≤ j ≤ m} for which g ′ (a) = h ′ (a) for any h ′ ∈ H a and g ′ ∈ G a . Denote by H C a and G C a the complementary subsets of monomials of H and G respectively, i.e., H C a = H \ H a and G C a = G \ G a . Then, for any h ′ ∈ H a and h ′′ ∈ H C a we have i.e., h ′ (a) ≥ h ′′ (a). We can write these last relations equivalently as 1 ∔ h ′′ (a) h ′ (a) = 1 for all h ′ ∈ H a and all h ′′ ∈ H C a . Similarly, for any g ′ ∈ G a and g ′′ ∈ G C a we have that g ′ (a) ∔ g ′′ (a) = g ′ (a). We can write these last relations equivalently as 1 ∔ g ′′ (a) g ′ (a) = 1 for all g ′ ∈ G a and all g ′′ ∈ G C a . Thus for any such a we obtain the following relations:

4)
Varying a ∈ Skel(f ), evidently there are only finitely many possibilities for relations in (13.3) and (13.4) as there are only finitely many monomials h i and g j comprising f .
Any set of relations comprised of the relations in (13.3) and (13.4), which will hereby be denoted by θ 1 and θ 2 , corresponds to a kernel generated by the corresponding elements The finite collection of pairs (θ 1 (i), θ 2 (i)) for i = 1, ..., s, is formed by considering all points a in Skel(f ).
Moreover, every point admitting the relations in (θ 1 (i), θ 2 (i)) for any i ∈ {1, ..., s} is in Skel(f ). Thus this collection supplies a complete covering of Skel(f ). Denote by K i , for i = 1, ..., s, the kernel generated by the elements of θ 1 (i) and θ 2 (i). Then since In essence s i=1 K i provides a local description of f in a neighborhood of is skeleton. We now proceed to supply an insight of the structure of the kernel f to put the construction above into a broader context.
In Construction 13.4.1, we used the skeleton of f to construct s i=1 K i . Considering all points a in R n might add some regions, complementary to the regions defined by (13.4) in θ 2 (i) for i = 1, ..., s, over which h ′ g ′ = 1, ∀h ′ ∈ H a , ∀g ′ ∈ G a , i.e., regions over which the dominating monomials never meet. Continuing the construction above using a ∈ R n \ Skelf similarly produces a finite collection of, say t ∈ Z ≥0 , kernels generated by elements from (13.4) and their complementary order fractions and by elements of the form (13.3) (where now h ′ g ′ = 1 over the considered region) . A principal kernel N j = q j 1 ≤ j ≤ t, of this complementary set of kernels has the property that Skel(N j ) = ∅, thus by Corollary 5.1.8, N j is bounded from below.
As there are finitely many such kernels there exists γ ∈ R, γ > 1 and small enough, such that |q j | ∧ γ = γ for j = 1, ..., t. Thus t j=1 N j is bounded from below and thus by Remark 5.1.3 we have that t j=1 N j ⊇ R .
As now the extension coincides with f over all of R n and as R is divisible, we have In view of the last discussion, we see that intersecting a principal kernel f with R 'chops off' all of its comprising bounded from below kernels (the N j 's above). This way it eliminates ambiguity in the kernel corresponding to Skel(f ).
Finally we note that if Skel(f ) = ∅ then f = t j=1 N j for appropriate kernels N j and f ∩ R = R .
A few notes concerning the construction: Remark 13.4.2.
1. If K 1 and K 2 are such that , then the sets of HP-fractions θ 1 of K 1 and of K 2 are not equal (though one may contain the other), for otherwise they would be combined via the construction to form a single kernel.

Let
. By Corollary 9.3.9, for any generator f ′ of f ∩ R we have .., s. In particular, Skel(k ′ i ) = Skel(|k i | ∧ |α|) = Skel(k i ). Thus the above construction of the K i kernels is independent of the choice of the generator as it is totally defined by the fragments Skel(k i ) of the skeleton Skel(f ).
We now provide two examples for the construction introduced above. We make use of the notation above for the different types of kernels involved in the construction.
Geometrically R 1,1 is a strip containing the axis x = 1 and R 2,1 is its complementary region. The restriction of f to R 1,1 gives it the form |x| while restricting to R 2,1 , f equals α. Furthermore, we see that over R 2,1 , f is bounded from below by α. Omitting N 2,1 we still have Skel(f ) = Skel(R 1,1 · L 1,1 ) though R 1,1 · L 1,1 ⊃ f and equality do not hold. Intersecting f with R leaves f intact while (R 1,1 Example 13.4.4. Let f = |x ∔ 1| ∧ α ∈ R(x, y) for some α > 1 in R. First note that since x∔ 1 ≥ 1 we have that |x∔ 1| = x∔ 1, allowing us to rewrite f as (x∔ 1)∧ α. Then The order relation α ≤ x translates to the relation α ∔ x = x or equivalently to αx −1 ∔ 1 = 1. Over the region defined by the last relation, we have As can be easily seen from examples 13.4.3 and 13.4.4 by substituting any HP-fraction for x and any order fraction for x ∔ 1 , the intersection of a kernel K = L · R = L i · O j (L i and O j are HP-kernels and order kernels respectively) with R yields where R ′ , R i and R ′ j are region kernels. R i c and R ′ j c are R i 's and R ′ j 's complementary region kernels respectively. N i , M j and N are bounded from below kernels. Note that the O ′ j s involve the R i s, the R ′ j s and the O j s, while N is derived from the bounded from below kernels, namely the N i s and M j s. Also note that intersecting with R keeps the HS-kernel unchanged in the new decomposition.
As the bounded from below kernels, the N j s in (13.5), do not affect Skel(f ) we leave them aside for the time being and proceed to study the structure of the kernels K i and their corresponding skeletons.
Let K be one of the above kernels K i . First note that every element of the set generating K which is specified above, is either an HP-fraction (a nonconstant Laurent monomial) of the form h ′ g ′ , or an order element of the form 1 ∔ h ′′ h ′ (or 1 ∔ g ′′ g ′ ). Let L i with i = 1, .., u and O j with j = 1, ..., v, be the kernels generated by each of the HP-kernels and the order kernels respectively. Then we can write where L = u i=1 L i is an HS-kernel and R = v j=1 O j is a region kernel. note that by the assumption of the construction above Skel(K) = ∅ (as there is at least one point of the skeleton used for constructing it). Moreover, there are no distinct HS-kernels M 1 and M 2 such that L = M 1 ∩M 2 for otherwise the construction would have produced two distinct kernels, one with M 1 as its HS-kernel and the other with M 2 as its HS-kernel instead of producing K in the first place. Definition 13.4.6. A principal kernel K ∈ PCon(R(x 1 , ..., x n )) is said to be an HO-kernel if it is generated by an HO-fraction.
Remark 13.4.7. A kernel K ∈ PCon(R(x 1 , ..., x n )) is an HO-kernel if and only if K = L · R where R is a region kernel and L is an HS-kernel.
Proof. If K is an HO-kernel, then K = f where f = l(f ) ∔ o(f ) is an HO-fraction.
is an HS-kernel and R = o(f ) is a region kernel. Conversely, taking the HO-fraction f = l ∔ r where l is an HS-fraction generating L and r is a region fraction generating R, we get that f = l ∔ r = l · r = L · R = K.
Remark 13.4.8. Let K = L · R be an HO-kernel with R a region kernel and L an HS-kernel. By Remarks 13.1.14 and 13.1.28, we have that L = u i=1 L i for some HPkernels L 1 , ..., L u and R = v j=1 O j for some order kernels O 1 , ..., O v . Thus K is of the form where u ∈ Z ≥0 , v ∈ N, and O 1 , ..., O v are order kernels. Note that every region kernel and every HS-kernel are by definition an HO-kernel, taking u = 0 for a region-kernel and v = 1 with O 1 = 1 ∔ a where L = a is the HS-kernel.
Remark 13.4.9. Let K 1 and K 2 be region-kernels (respectively HS-kernels) such that Then K 1 · K 2 is a region-kernel (respectively HS-kernel).
Consequently, if K 1 and K 2 are HO-kernels such that K 1 · K 2 ∩ R = {1} , then K 1 · K 2 is an HO-kernel. Indeed, the assertions follow from the decomposition with the appropriate u s , v s taken for s = 1, 2.
By the above discussion we have where each K i is a product of an HS-kernel and a region kernel 8) while each N i is a product of bounded from below kernels and (complementary) region kernels. For f ∈ PCon ( R ), the N j can be replaced by R without affecting the resulting kernel.
• If f is an HS-kernel, then the decomposition degenerates to f = K 1 with • f is an irregular kernel if and only if there exists some i 0 ∈ {1, ..., s} such that • f is a regular kernel if and only if K i is comprised of at least one HP-kernel, for every i = 1, ..., s.
Proof. The last three assertions are direct consequences of the construction above, namely, if f is either an HS-kernel or a region kernel, f is already in the form of its decomposition. The fourth is equivalent to the third.
If f is an HS-kernel then by Remark 13.1.14, f = t j=1 L j where L j is an HP-kernel for each j. Write f = h i g j with h i , g j ∈ R[x 1 , ..., x n ] monomials. If f is irregular, then by definition of irregularity we have some i 0 and j 0 for which h i 0 = g j 0 where (g j 0 =)h i 0 > h i , g j for every i = i 0 and j = j 0 , at some neighborhood of a point a ∈ R n . The kernel corresponding to (the closure) of this region has its relation (13. 3) degenerating to 1 = 1 as h i 0 g j 0 = 1 over the region, thus is given only by its order relations of (13.4).
Definition 13.4.11. We call the decomposition given in Theorem 13.4.10 of a principal kernel f ∈ PCon(R(x 1 , ..., x n )) the HO-decomposition of f . In the special case where f ∈ PCon( R ), all bounded from below terms of the intersection are equal to R . Example 13.4.14. Consider the kernel f where f = x y+1 ∈ R(x 1 , ..., x n ). The points on the skeleton of f define three distinct HS-kernels: x y (corresponding to the equality x = y) over the region {y ≥ 1} which is defined by the region kernel 1+y −1 , x = x 1 (corresponding to x = 1) over the region {y ≤ 1} which is defined by the region kernel 1 + y , and |x| + |y| (corresponding to the point defined by x = 1 and x = y). Thus by Construction 13.4.1 and Note that the third component of the decomposition (i.e., the HS-kernel |x| + |y| ) can be omitted without effecting Skel(f ).
The decomposition is shown (in logarithmic scale) in Figure 13.2 where the first two components are the rays beginning at the origin and the third component is the origin itself.
13.5 The lattice generated by regular corner-integral principal kernels Recall Remark 13.1.31 which states that the principal kernel f · g 1 · · · · · g k is regular, for any HP-kernel f = 1 and order kernels g 1 , ..., g k .
Corollary 13.5.1. Let K ∈ PCon( R ) and let be the decomposition of K, as given in (13.6), where L 1 , ...., L u are some HP-kernels and O 1 , ..., O v are some order kernels. If u = 0, i.e., L = 1 then K is regular.
Proof. Indeed, K = u i=2 L i · (L 1 · v j=1 O j ) ∩ R . By Remark 13.1.31, we have that (L 1 · v j=1 O j ) is regular as L 1 is regular as an HP-kernel. Thus since a product of regular kernels is regular and since intersection with R does not affect regularity, we have that K is a regular kernel.
Theorem 13.5.2. The lattice generated by principal corner integral kernels in PCon (R) is the lattice of principal kernels PCon(R) and the lattice generated by regular principal corner integral kernels is the lattice of regular principal kernels.
Proof. Let f be a principal kernel and let O l,i be its HO-decomposition. By Lemma 13.1.29, each HP-kernel L j,i and each order kernel O l,i are corner integral. Thus f as a finite product of principal corner integral kernels is in the lattice generated by principal corner-integral kernels. As any principal corner integral kernel is in particular principal, the lattice of principal kernels contains the lattice generated by principal corner-integral kernels proving the first assertion. As for the second assertion, if f is regular, then by Theorem 13.4.10, for every 1 ≤ i ≤ s, we have that L 1,i = 1. Thus by Corollary 13.5.1 we have that each K i is a product of principal regular corner-integral kernels. Thus f is in the lattice generated by principal regular corner-integral kernels. As any regular corner integral kernel is in particular regular, the lattice of principal regular kernels contains the lattice generated by principal regular corner-integral kernels. Thus the second assertion holds.
Corollary 13.5.3. By Theorem 13.5.2, and the correspondence between principal (regular) corner-loci and principal (regular) corner-integral kernels of R , defined by composing the correspondence between the kernels with their principal skeletons K → Skel(K) introduced in Corollary 7.0.26 with the correspondence between these principal skeletons and their corresponding corner loci introduced in Proposition 10.2.18, we have that the lattice of (regular) finitely generated corner loci corresponds to the lattices of principal (regular) kernels of R .
Corollary 13.5.4. By Corollary 13.5.3, we have that supertropical varieties correspond to principal skeletons and kernels while tropical varieties correspond to regular principal skeletons and kernels.

Convexity degree and hyperdimension
In this section, K is a semifield which is an affine extension of the bipotent semifield R, i.e., K is of the form R(x 1 , ..., x n )/L for some kernel L ∈ Con(R(x 1 , ..., x n )) (see Remark 2.3.23). In particular K is idempotent.
Proposition 13.6.1. Let S be an idempotent semifield. Let M be a kernel of S such that M = K 1 ∩ K 2 ∩ · · · ∩ K t for distinct kernels K i of S. Then S/M is subdirectly reducible and Proof. Consider the quotient semifieldS = S/M . Let φ : S →S be the quotient map.
Denote byK i = φ(K i ) = K i /M the images of K 1 , ..., K n under φ which are kernels of S. Since M = K 1 ∩ K 2 ∩ · · · ∩ K t , we have thatK 1 ∩K 2 ∩ · · · ∩K t = {1} inS (note that by Corollary 2.2.53 there is a lattice isomorphism between the kernels ofS and the kernels of S that contain M ).
Thus, by Remark 3.1.12, we haveS is subdirectly reducible and as S/K i = (S/M )/(K i /M ) ∼ = S/K i (by the second isomorphism theorem 2.2.52) we have Corollary 13.6.2. In particular, Proposition 13.6.1 applies to the idempotent semifield H(x 1 , ..., x n ) where H is a bipotent semifield (or any of its kernels considered as a semifield) and to any principal kernel M = f ∈ PCon(H(x 1 , ..., x n )) such that M = K 1 ∩ K 2 ∩ · · · ∩ K t for some K i ∈ PCon (H(x 1 , ..., x n )).
Let f ∈ R be a principal kernel and let f = s i=1 K i , where is its (full) HO-decomposition; i.e., for each 1 ≤ i ≤ s R i ∈ PCon(R(x 1 , ..., x n )) is a region kernel and L i ∈ PCon(R(x 1 , ..., x n )) is either an HS-kernel or bounded from below (in which case L ′ i = R ). Then by Corollary 13.6.2, we have the subdirect decomposition where t ≤ s is the number of kernels K i for which L ′ i = R (for otherwise R /K i = {1} and can be omitted from the subdirect product).
The last discussion motivates us to study the semifields R /K i as building blocks for the algebraic structure of the quotient semifield R / f which in turn is the coordinate semifield corresponding to the skeleton Skel(f ).
Examples 13.6.3 and 13.6.4 demonstrate that the lattice of principal kernels PCon(R(x 1 , ..., x n )) (resp. PCon( R )) is too rich to define reducibility or dimensionality. Moreover, these examples suggest that this richness is caused by order kernels. This motivates us to consider Θ-reducibility for some sublattice of kernels Θ ⊂ PCon(R(x 1 , ..., x n )) (resp. Θ ⊂ PCon( R )). There are various families of kernels that one can utilize to define the notions of reducibility, dimensionality, etc.
We consider the sublattice generated by HP-kernels. Our choice is made due to its connection to the (local) dimension of the linear spaces (in logarithmic scale) defined by the skeleton corresponding to a kernel. Namely, HP-kernels, and more generally HS-kernels, define affine subspaces in R n (see 13.2). Definition 13.6.5. Let K be a semifield as defined in the beginning of this section. Define S(K) to be the set elements of K generated by the collection {|f | : f ∈ K ; f is an HP-fraction } with respect to the operations ∧ and ∔ (equivalently ∨). Since K is a semifield it is closed with respect to ∧ and ∨ and | · |, thus S(K) ⊂ K. By definition, all HS-fractions in K are elements of S(K). Define Γ(K) = S(K)/ ∼ K .
Namely, for f, g ∈ Γ(K), f ∼ K g if and only if f = g in PCon(K). Note that 1 ∈ Γ(K).
Let Ω(K) ⊂ PCon(K) be the lattice of kernels generated by the collection of all HP-kernels of K, i.e., every element f ∈ Ω(K) is obtained by a finite intersections and products of HP-kernels.
By definition, every HS-kernel in PCon(K) is an element of Ω(K) as it is a product of finite set of HP-kernels. Note that 1 ∈ Ω(K).
Note that f = |f | and g = |g| by the definition of Γ. If f, g ∈ Γ such that ρ(f ) = ρ(g), then by definition f = g and thus f ∼ K g. So, ρ is injective. Finally, ρ is onto by the definition of HP-kernels.
As Ω(K) is a sublattice of kernels of PCon(K), we have the notion of Ω(K)-irreducible kernel as developed in Subsection 9.2 concerning 'reducibility of principal kernels and skeletons' (taking Θ = Ω(K)). Namely f ∈ Ω(K) is reducible if there exist some g , h ∈ Ω(K) such that f ⊇ g ∩ h while f ⊇ g and f ⊇ h .
Lemma 13.6.7. By the construction of Ω(K), the condition stated above is equivalent to the condition f = g ∩ h while f = g and f = h .
Proof. Assume f admits the stated condition. If f ⊇ g ∩ h , then f = f · f = g · f ∩ h · f . Thus f = g · f or f = h · f and so f ⊇ g or f ⊇ g .
The converse is obvious.
Note 13.6.8. For the rest of this section, we refer to Ω(K)-irreducibility as irreducibility.
Definition 13.6.9. Define the irreducible hyperspace spectrum of K, HSpec(K), to be the family of irreducible kernels in Ω(K).
Remark 13.6.10. HSpec(K) is the family of HS-kernels in Ω(K) which is exactly the family of HS-kernels of K.
Definition 13.6.11. A chain P 0 ⊂ P 1 ⊂ · · · ⊂ P t in HSpec(K) means an ascending chain of HS-kernels of K, and is said to have length t. An HS-kernel P has height t (denoted hgt(P ) = t) if there is a chain of length t in HSpec(K) terminating at P , but no chain of length t + 1 terminates at P .
Remark 13.6.12. Let L be a kernel in PCon(K). Consider the quotient homomorphism φ L : K → K/L. As the image of a principal kernel is the principal kernel generated by the image of any of its generators, we have that for an HP-kernel f we have φ L ( f ) = φ L (f ) . Choosing f to be in canonical HP-fraction, since φ L is an R homomorphism, we have that φ L (f ) is a nontrivial HP-kernel in K/L if and only if φ L (f ) ∈ R. Thus the set of HP-kernels of K mapped to HP-kernels of K/L is (13.10) Since φ L is an R-homomorphism and onto, we have that one of the preimages of a canonical HP-fraction g ∈ K/L (represented by g) is g ∈ K itself. As φ L is an R-homomorphism it respects ∨, ∧ and | · | thus φ L (Γ(K)) = Γ(K/L).
In fact by Corollary 2.7.11, we have a correspondence identifying HSpec(K/L) with the subset of HSpec(K) which consists of all HS-kernels P of K such that P · R ⊇ L · R . Moreover, by the same considerations and since ∧ is preserved under a homomorphism we have that the above correspondence extends to a correspondence identifying Ω(K/L) with the subset (13.10) of Ω(K). Under this correspondence, the maximal (HS) kernels of K/L correspond to maximal (HS) kernels of K and reducible kernels of K/L correspond to reducible kernels of K. Indeed, the latter assertion is obvious since ∧ is preserved under a homomorphism. For the former assertion, by the second isomorphism theorem (K/L)/(P/L) ∼ = K/P so simplicity of the quotients is preserved. Thus so is maximality of P and P/L.
Definition 13.6.13. The Hyperdimension of K, written HdimK (if it exists), is the maximal height of the HS-kernels in K.
Definition 13.6.14. Let A ⊂ HS(K) be any set of HS-fractions and let f ∈ HS(K). Then f is said to be R-convexly dependent on A if f ∈ {g : g ∈ A} · R , (13.11) otherwise f is said to be R-convexly-independent of A. A subset A ⊂ HS(K) is said to be R-convexly independent if for every a ∈ A, a is R-convexly independent of A \ {a} over R. Note that by assuming g ∈ K \ {1} for some g ∈ A the condition in (13.11) simplifies to f ∈ {g : g ∈ A} . Indeed, under this last assumption we have that R ⊆ {g : g ∈ A} and so {g : g ∈ A} · R = {g : g ∈ A} .
Remark 13.6.16. By the definition, we have that an HS-fraction f is R-convexly dependent on {g 1 , ..., g t } ⊂ HS(K) if and only if where α is any element of K \ {1}.
Proof. By Remark 13.6.29, f i is dependent on {f i,j : 1 ≤ j ≤ t i } ⊂ S for each 1 ≤ i ≤ m, thus ConSpan(S) = ConSpan(V ). By Corollary 13.6.24, S contains a maximal R-convexly independent subset S 0 such that ConSpan(S 0 ) = ConSpan(S). By Lemma 13.6.30 for each i = 1, ..., m, there exists some HP-fraction g i = f i,j i such that g i is R-convexly independent of V \ {f i }, for otherwise f i would be R-convexly dependent on V \ {f i }, thus {g 1 , ..., g m } ⊂ S is R-convexly independent, so |S 0 | ≥ m = |V |. Now that our restriction to HP-fractions is justified, we move forward with our construction.
Remark 13.6.32. Let K ∈ K be an HS-kernel. Consider the following set HP(K) = {f ∈ K : f is an HP-fraction } .
Definition 13.6.33. Let K ∈ K be an HS-kernel. In the notation of Remark 13.6.32, we define the convexity degree of ConSpan(K), condeg(K) to be s, the number of elements in a basis B.
The last equality is due to the fact that |h i | k ≥ 1 so that 1 is absorbed. The same arguments applied to the left hand side of equation (13.13) yields that |h 1 | ∔ g ≤ |h 2 | k ∔ · · · ∔ |h t+1 | k ∔ g k . (13.14) Assume on the contrary that h 1 is R-convexly independent of {h 2 , ..., h t }. Then Thus for any m ∈ N there exists some x m ∈ R n such that Thus by equation (13.14) and the last observation we get that i.e., there exists some fixed k ∈ N such that for any m ∈ N, For m > k, since |γ| k ≤ |γ| m for any γ ∈ R, we get that t i=2 |h i (x m )| m ≥ t i=2 |h i (x m )| k . Write g k (x) = g(1)g ′ (x). Since g k is an HP-kernel, g(1) is the constant coefficient of g and g ′ is a Laurent monomial with coefficient 1. Now, by the way x m were chosen we have that t i=2 |h i (x m )| > 1 and t i=2 |h i (x m )| m < g(x m ) k , and thus for sufficiently large m 0 we have that g(1) < g ′ (x m 0 ). Since g ′ is a Laurent monomial with coefficient 1, we have that g ′ (x −1 m ) = g ′ (x m ) −1 so g k (x −1 m 0 ) = g(1)g ′ (x −1 m 0 ) = g(1)g ′ (x m 0 ) −1 < 1. Thus by (13.15) we must have t i=2 |h i (x −1 m 0 )| m < t i=2 |h i (x −1 m 0 )| k . A contradiction. Proof. The 'if' part of the assertion follows Proposition 13.6.36. As R = m i=1 O i for some order -kernels {O i } m i=1 , the 'only if' part follows from Lemma 13.6.37 by applying it repeatedly to each of the O i 's comprising R.
Thus equality holds.
Remark 13.6.41. Let R be a region kernel of K and let A = { g : g · R ⊇ R · R }.
In view Remark 13.6.12 and of Proposition 13.6.38, we have that condeg(K/R) = condeg(A). As L a ⊇ A for any a ∈ Skel(R) = ∅, we have that condeg(A) = condeg(K).
We can always choose a basis for HP(K/R) containing a basis for HP(L · R/R). Taking L instead of K in Remark 13.6.41, we have that condeg(L · R/R) = condeg(φ R (L)).
Consider the following descending chain of HS-kernels where Skel(K) = ∅ Proof.
Thus the inclusions of the chain (13.16) are strict, i.e., it is strictly descending. On the other hand if h u is R-convexly dependent on {h 1 , ...., h u−1 } then L u = h u ⊆ u−1 i=1 L i · R . Assume L u = h u ⊆ u−1 i=1 L i , then R ⊆ u i=1 L i implying that u i=1 L i is not an HS-kernel. Thus L u = h u ⊆ u−1 i=1 L i and the chain is not strictly descending.
Proposition 13.6.44. If L ∈ HSpec(K), then hgt(L) = condeg(L). Moreover, every factor of a descending chain of maximal length is an HP-kernel.
Proof. By Proposition 13.6.43, we have that the maximal length of a chain of HS-kernels descending from an HS-kernel L equals the number of elements in a basis of ConSpan(L); thus we have that the chain is of unique length condeg(K), i.e., hgt(L) = condeg(L). Moreover, by Theorem 2.2.51 (2) we have that So the homomorphic image of the HPkernel L j under the quotient map R(x 1 , ..., x n ) → R(x 1 , ..., x n )/(L j ∩ ( j−1 i=1 L i )) is an HP-kernel. Thus every factor of the chain is an HP-kernel.
Proof. A consequence of Definition 13.6.13 and Proposition 13.6.44.
Let K ∈ Con(R(x 1 , ..., x n )). Then Since K itself is an idempotent semifield, we have that As we have seen for every principal regular kernel f ∈ (R(x 1 , ..., x n )), there exists an explicitly formulated cover of regions {R 1,1 , ..., R 1,s , R 2,1 , ..., R 2,t where K i = L i · R 1,i for i = 1, ..., s and appropriate HS-kernels L i and N j = B j · R 2,j for j = 1, ..., t and appropriate bounded from below kernels B j . If f ∈ PCon( R ), as we have shown, then B j = R for every j = 1, ..., t. Note that over the different regions in R n , corresponding to the region kernels R i,j , f is locally represented by distinct elements of HSpec(R(x 1 , ..., x n )) (HS-fractions). In fact the regions themselves are defined such that the local HS-representation of f will stay invariant over each. Thus the R i,j 's, defining the partition of the space, are uniquely determined as the minimal set of regions over each of which f comes from a unique HS-kernel.