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Hypervaluations on hyperfields and ordered canonical hypergroups

We study the concept of hypervaluations on hyperfields. In particular, we show that any hypervaluation from a hyperfield onto an ordered canonical hypergroup is the composition of a hypervaluation onto an ordered abelian group (which induces the same valuation hyperring) and an order preserving homomorphism of hypergroups.


Introduction
The theory of hyperrings and hyperfields has its origins in the paper of Krasner [3]. More recently, several authors studied their theory from various points of view. For instance, J. Jun, in [2], studies algebraic geometry over hyperrings and gives the definition of hyperideals in a hyperring. M. Marshall in [6] studies the definitions of orderings and of positive cones in hyperfields. It was already mentioned in that paper that the two concepts are not equivalent as in the classical theory of those objects. In fact, we show in Example 2.15 that the natural generalization of the construction which in the classical case leads to this equivalence, does not work in the case of hyperfields.
In [1] B. Davvaz and A. Salasi deal with hypervaluations on a hyperring onto an ordered abelian group, and also J. Lee in [5] works with valued hyperfields. These two papers present different approaches in defining a hypervaluation on a hyperfield, which both appear to be interesting and well chosen.
In this paper we study another possibility, introduced by Kh. Mirdar Harijani and S. M. Anvariyeh in A hypervaluation of a hyperfield onto a totally ordered canonical hypergroup, Studia Scientiarum Mathematicarum Hungarica 52 (1), 87-101 (2015). What they propose is a generalization of the definition of Davvaz and Salasi where the value set is allowed to be an ordered canonical hypergroup (see Definition 2.10). Here the domain is always assumed to be a hyperfield. We show in our main result, that even though this definition is proper, i.e., nontrivial examples can be found, it is not of much interest since such a hypervaluation can always be decomposed into an order preserving homomorphism of hypergroups and a hypervaluation onto an ordered abelian group (in the sense of Davvaz and Salasi) which, moreover, induces the same valuation hyperring.
In the paper of Mirdar Harijani and Anvariyeh, a lot of constructions are proposed without details; it turns out that they are not always possible to carry out. In the present article we also provide counterexamples to justify this last assertion.

Ordered Canonical Hypergroups
Definition 2.1. Let H be a nonempty set and P * (H) the family of nonempty subsets of H. A hyperoperation * is a function which associates with every pair (x, y) ∈ H × H an element of P * (H), denoted by x * y.
A hypergroupoid is a nonempty set H with a hyperoperation * : In 1934 the concept of a hypergroup was defined by F. Marty in [7] to be a nonempty set H with an associative hyperoperation (see Definition 2.2 below) such that x * H = H * x = H for all x ∈ H. A special class of hypergroups, which will be of interest for us, is the following: where (H, * ) is a hypergroupoid and e is an element of H such that the following axioms hold: (H1) the hyperoperation * is associative, i.e., (x * y) * z = x * (y * z) for all x, y, z ∈ H, (H2) x * y = y * x for all x, y ∈ H, (H3) for every x ∈ H there exists a unique x ′ ∈ H such that e ∈ x * x ′ (the element x ′ will be denoted by x −1 ), (H4) z ∈ x * y implies y ∈ x −1 * z for all x, y, z ∈ H. Remark 2.3. A canonical hypergroup is a hypergroup in the sense of Marty. Fix a ∈ H and take Remark 2.4. In [8] the definition of a canonical hypergroup also requires explicitly that x * e = {x} for all x ∈ H. However, we note that this axiom follows from (H3) and (H4). Indeed, suppose that y ∈ x * e for some x, y ∈ H. Then e ∈ x −1 * y by (H4). Now y = x follows from the uniqueness required in (H3).
Remark 2.5. Note that an abelian group G is not a priori a hypergroup, because the operation on G is not a hyperoperation, as it takes values in G and not in P * (G). But it can be turned into a hypergroup by setting a * b := {ab}. In other words, we can turn an abelian group into a hypergroup by identifying each element a of G with the singleton {a}.
The next example can be found in [4].
Example 2.7. Let R be a ring and G a normal subgroup of its multiplicative semigroup. Consider the following equivalence relation ∼ on R: a ∼ b if and only if there exist g, h ∈ G s.t. ag = bh. The equivalence class of a ∈ R is It is possible to define a hyperoperation on R/G in the following way: Then (R/G, +, {0 R }) is a canonical hypergroup. Indeed, the associative law follows from the same law in R, as well as commutativity. The unique inverse of aG is (−a)G. Indeed, h ∈ G}, then there exist g, h ∈ G such that 0 R = ag +bh. Multiplying by g −1 , we obtain that −a = bhg −1 and so (−a)G = bG.
Assume now that cG ∈ aG + bG. We wish to show that bG ∈ (−a)G + cG. We have Remark 2.8. We note that the sign hypergroup can be obtained as a quotient in the way described in the previous example. Take R = R and G =Ṙ 2 , whereṘ 2 denotes the set of non-zero squares in R. The result follows from the fact that every non-zero real number is either a square or the opposite of a square.
In their paper Mirdar Harijani and Anvariyeh consider the set Z/N = {aN | a ∈ Z} with the hyperoperation defined as follows: This construction does not give a canonical hypergroup in the sense of Definition 2.2, as is shown in the next example. The reason is that N is not a multiplicative subgroup of the multiplicative semigroup of Z. Example 2.9. Consider the tuple (Z/N, * , 0N). We observe that 0N ∈ 1N * −kN for every k ∈ N. Hence axiom (H3) is not fulfilled. Definition 2.10. An ordered canonical hypergroup is a tuple (H, * , e, ≤), where (H, * , e) is a canonical hypergroup and ≤ is a partial order such that then we call ≤ an ordering or a linear order relation on H.
Remark 2.11. One should not use "≤" on the subsets as the relation will not be antisymmetric. For example if H contains a smallest element b, then any two subsets A, B, which both contain b will satisfy A ր B and B ր A without necessarily being equal.
Remark 2.12. It should be noted that some authors in defining A ր B require that for all a ∈ A there exists b ∈ B and for all b ∈ B there exists a ∈ A such that a ≤ b. Others instead require only that for all a ∈ A there exists b ∈ B such that a ≤ b. The relations between all these definitions still have to be investigated. However, in what follows we will just use the concept introduced in Definition 2.10.
Lemma 2.13. Let (H, * , e, ≤) be an ordered canonical hypergroup. Take a, b, x, y ∈ H and x > e and y ≥ e, then b > e for all b ∈ x * y.
Proof. 1): This follows from the definition of "ր" and the fact that {a} contains only a.
2): By condition (1), which implies that x = e in contradiction to our assumption that x > e.
3): If x ≥ e and y ≥ e, then {y} = e * y ր x * y, so e ≤ y ≤ b for every b ∈ x * y by part 1).

4): If
x > e and y ≥ e, then b ≥ e for all b ∈ x * y by part 3). However, e / ∈ x * y since otherwise, y = x −1 which by part 2) yields that y < e, in contradiction to our assumption that y ≥ e. Hence b = e and consequently, b > e.
In [6] the definition of a positive cone is given for hyperfields. Let us now define this concept for canonical hypergroups.
Definition 2.14. A subset P of a canonical hypergroup (H, * , e) is called a positive cone if the following axioms hold: In the theory of ordered abelian groups, the existence of a positive cone is equivalent to the existence of an ordering and there is a one to one correspondence between them. It was already mentioned in [6] that this, in general, is no more true in the case of hyperfields, and the argument holds for hypergroups as well. Mirdar Harijani and Anvariyeh claim that one can always construct an ordering from a positive cone by setting: (2) x ≤ y ⇐⇒ (y * x −1 ) ∩ P = ∅.
In the following example we show that this construction is not always possible.
Example 2.15. Take H := Q/Q 2 , whereQ 2 denotes the set of nonzero squares in Q. We see that H is a canonical hypergroup with the hyperoperation defined as: (see also Example 2.7). Observe that the set P = Q + /Q 2 ∪ {0} fulfils the conditions of the definition of a positive cone. However, the relation defined in (2) is not an ordering on H. Indeed, Clearly, 2Q 2 = 3Q 2 , so the relation ≤ is not antisymmetric.
However, there exist hypergroups in which positive cone and orderings behave as in the classical case. The simplest example is the sign hypergroup: We define an order relation ≤ on H as follows: −1 ≤ 0 ≤ 1. We leave it to the reader to show that ≤ is a linear order on H as in Definition 2.10. This ordering corresponds to P in the way described in (2). Indeed, , so in every case we obtain a nonempty intersection.
The following example is the original example of a quotient hyperring (see [4]).
Example 3.2. Let R be a commutative ring with 1, G a normal subgroup of its multiplicative semigroup and recall the notations introduced in Example 2.7. One may define multiplication in R/G as aG · bG := (ab)G. Then (R/G, +, ·, {0}, 1 R G) is a hyperring and if R is a field, then it is a hyperfield. We have to show that (R3) holds. Take a, b, c ∈ R and suppose that xG ∈ aG · (bG + cG). Then there exist g, h ∈ G such that x = a(bg + ch) = abg + ach, where we have used the distributivity law in R. We obtain This shows that aG · (bG + cG) ⊆ aGbG + aGcG. To show the converse inclusion we note that xG ∈ aGbG + aGcG = (ab)G + (ac)G.
If R is a field, then for every a ∈ R we have that a −1 G = (aG) −1 . Indeed, by definition aG · a −1 G = (a · a −1 )G = 1 R G. Example 3.3. As we have already noted in Remark 2.8 that the sign hypergroup H can be seen as a quotient R/Ṙ 2 . Since R is a field, we obtain that (H, +, ·, 0, 1) is a hyperfield, where we now denote * by + and · behaves as follows: In their paper Mirdar Harijani and Anvariyeh, in an attempt to generalize the construction of formal power series, argue that the set X of order preserving mappings f : H → K such that the support of f is finite, where H is a ordered canonical hypergroup and K an ordered hyperfield, is a hyperdomain, i.e., a hyperring without zero divisors, with the operations defined as follows: First of all we note that X is not even a hyperring since if f is non-constant and order preserving, then −f is not order preserving. In addition, if K is a hyperfield which is not a field, then the multiplication (f g)(x) defined in (3) is a subset of K and not an element of K as the definition of hyperring would require. Hence we do not obtain a hyperring. Finally, even if we do not restrict to order preserving mappings and assume K to be a field, we show, in the next example, that the multiplication defined in (3) is not associative.

Hypervaluations
As it was mentioned before, there are several approaches to the definition of a hypervaluation on a hyperfield. We now wish to investigate the following.
The following is a nontrivial example of a hypervaluation onto an ordered canonical hypergroup.
Example 4.2. Let K be a field and Γ an ordered abelian group. Assume that a classical valuation v : K → Γ ∪ {∞} is given. We now define a hypervaluation w from K onto the sign hypergroup {−1, 0, 1}.
Definition 4.4. Let (R, +, ·, 0, 1) be a hyperring. An element x ∈ R is called a unit of R if there exists y ∈ R such that x · y = 1.
Definition 4.5. Let F be a hyperfield and R ⊆ F a hyperring with respect to the hyperaddition and multiplication of F . If for every x ∈ F either x ∈ R or x −1 ∈ R, then R is called a valuation hyperring.
Definition 4.6. Let R be a hyperring.
(1) A nonempty subset I ⊆ R is a hyperideal if for all a, b ∈ I and for all r ∈ R we have a + b ⊆ I, −a ∈ I and ar ∈ I. (2) A hyperideal I R is maximal if I satisfies the following property: if J ⊆ R is a hyperideal of R such that I J, then J = R.
In the paper of Mirdar Harijani and Anvariyeh one can find the first three statements of the following proposition. For the sake of completeness we rewrite the proof and complete it with details where needed. For our purposes, we also add a fourth statement. 1) The set O w = {x ∈ F | w(x) ≥ e} is a valuation hyperring.
2) The set U w = {x ∈ F | w(x) = e} is a group under the multiplication of the hyperfield F and consists of all units of O w .
3) The set m w = {x ∈ F | w(x) > e} is the unique maximal hyperideal of O w . 4) The quotient G := (F \ {0})/U w is an ordered abelian group with respect to the operation xU w · yU w = xyU w and the ordering xU Proof. 1) If x, y ∈ O w , then w(x), w(y) ≥ e, so by part 3) of Lemma 2.13 we obtain that for all b ∈ w(x) * w(y) we have b ≥ e. Since w(xy) ∈ w(x) * w(y), in particular w(xy) ≥ e, hence xy ∈ O w . The inclusion x + y ⊆ O w follows from (V4) and from the fact that min{w(x), w(y)} ≥ e. Indeed by (V4), z ∈ x + y =⇒ w(z) ≥ min{w(x), w(y)} ≥ e. To see that O w is a valuation hyperring, take x ∈ F \ O w . Then w(x) < e, which implies that w(x −1 ) = (w(x)) −1 > e by Lemma 4.3 and part 2) of Lemma 2.13. Thus, x −1 ∈ O w .
2) If x ∈ U w , then x −1 ∈ U w too. Indeed, w(x −1 ) = (w(x)) −1 = e. Hence the elements of U w are units of O w . To show the other inclusion, assume that x ∈ O w \ U w . Then w(x) > e, so w(x −1 ) < e by part 2) of Lemma 2.13, which means that 3) Let x, y ∈ m w . Then by (V4) we obtain that Thus, x + y ⊆ m w . From w(−x) = w(x) it follows that −x ∈ m w . If x ∈ m w and y ∈ O w , then w(x) > e and w(y) ≥ e, so by part 4) of Lemma 2.13 we obtain that for every b ∈ w(x) * w(y) we have b > e. Since by (V3) w(xy) ∈ w(x) * w(y), we conclude that xy ∈ m w . We have proved that m w is a hyperideal. It follows from part 2) and from the fact that if a hyperideal contains a unit, then is not proper, that m w is the unique maximal hyperideal of O w . 4) By part 2) U w is a subgroup of the abelian group (F \ {0}, ·, 1). So G is the quotient group with respect to this (normal) subgroup. The proof that ≤ is a linear order on G is exactly the same as in the classical case.
Note that there is no order preserving isomorphism between the ordered abelian group Γ and the sign hypergroup. But in the present case, as shown above, they define the same valuation (hyper)ring in K.
The next theorem states that any hypervaluation from a hyperfield onto an ordered canonical hypergroup is the composition of a hypervaluation onto an ordered abelian group (which induces the same valuation hyperring) and an order preserving homomorphism of hypergroups.
Proof. By part 4) of Proposition 4.7 we can consider the ordered abelian group G = (F \{0})/U w . Let h : G → H be the map h(xU w ) = w(x). We first show that h is well defined. Suppose that xU w = yU w . This is equivalent to xy −1 ∈ U w , which holds if and only if w(xy −1 ) = e. From (V3) and Lemma 4.3 we obtain that e ∈ w(x) * w(y −1 ) = w(x) * (w(y)) −1 and this means that w(x) = w(y) by the uniqueness required in axiom (H3). This proves that h is well defined. To show that h is a homomorphism of hypergroups, note that h(1U w ) = w(1) = e by Lemma 4.3. Take xU w , yU w ∈ G. According to Remark 2.5 we obtain that h(xU w · yU w ) = {h(xyU w )} = {w(xy)} ⊆ w(x) * w(y) = h(xU w ) * h(yU w ), which proves that h is a homomorphism of hypergroups. Moreover, assume that xU w ≤ yU w , which by definition is equivalent to e ≤ w(yx −1 ). From the latter it follows by condition (1) of Definition 2.10 that {w(x)} = e * w(x) ր w(yx −1 ) * w(x). Hence by part 1) of Lemma 2.13, w(x) ≤ b for every b ∈ w(yx −1 ) * w(x). As w(y) = w(yx −1 x) ∈ w(yx −1 ) * w(x), we obtain that w(x) ≤ w(y). This proves that h is order preserving.
We now define v : F → G ∪ {∞} as follows: The map v is a hypervaluation onto the ordered abelian group G (again, modulo the provision of Remark 2.5). The only axiom which needs justification is the fourth: if z ∈ x + y for some x, y ∈ F , then we wish to show that v(z) ≥ min{v(x), v(y)}. Assume, without loss of generality, that v(x) ≤ v(y), so that yx −1 ∈ O w . From z ∈ x + y it follows that This means that zx −1 ∈ yx −1 − 1 ⊆ O w , which proves that v(z) ≥ v(x). Finally, one can see that w = h • v. Indeed, if x ∈ F , then if x = 0, we have that h(v(x)) = h(xU w ) = w(x) by definition. If x = 0, the situation is clear once we set h(∞) := ∞. It is left to show that O v = O w . We observe that x ∈ O w if and only if 1U w ≤ xU w , which happens if and only if x ∈ O v . This completes the proof.