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Number Systems with Simplicity Hierarchies II

In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No, i.e. a subfield of No that is an initial subtree of No. In this sequel to [15], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. It is further shown that an initial subdomain of No is discrete if and only if it is an initial subdomain of No's canonical integer part Oz of omnifc integers. Finally, extending results of [15], the theories of divisible ordered abelian groups and real-closed ordered fields are shown to be the sole theories of ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of No.

numbers including −ω, ω/2, 1/ω, √ ω and e ω , to name only a few. This particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers-construed here as members of ordered fields-be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with global choice (NBG) [23], it may be said to contain "All Numbers Great and Small." In this respect, No bears much the same relation to ordered fields that the ordered field R of real numbers bears to Archimedean ordered fields ( [13], [15], [19]). In addition to its inclusive structure as an ordered field, No has a rich simplicity hierarchical ( or s-hierarchical) structure [15], [14], that depends upon its structure as a lexicographically ordered full binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No's structure as an ordered group and an ordered field, respectively, it being understood that x is simpler than y just in case x is a predecessor of y in the tree.
Among the striking s-hierarchical features of No that emerged from [15] is that much as the surreal numbers emerge from the empty set of surreal numbers by means of a transfinite recursion that provides an unfolding of the entire spectrum of numbers great and small (modulo the aforementioned provisos), the recursive process of defining No's arithmetic in turn provides an unfolding of the entire spectrum of ordered abelian groups (ordered fields) in such a way that an isomorphic copy of every such system either emerges as an initial subtree of No or is contained in a theoretically distinguished instance of such a system that does. In particular, it was shown that every divisible ordered abelian group (real-closed ordered field) is isomorphic to an initial subgroup (initial subfield) of No.
The divisible ordered abelian groups and real-closed ordered fields, however, do not exhaust the ordered abelian groups and ordered fields that are isomorphic to initial subgroups and subfields of No. For example, every 2divisible Archimedean ordered abelian group has an initial isomorphic copy in No, as does and every Archimedean ordered field [15,Theorem 8], and these groups and fields of course are not in general divisible or real-closed. In the case of ordered fields, more generally, in [15,Theorem 18] it was shown that: An ordered field is isomorphic to an initial subfield of No if and only if it is isomorphic to a truncation closed, cross sectional subfield of a power series field R((t Γ )) On where Γ is isomorphic to an initial subgroup of No.
The present paper is a sequel to [15]. Following some preliminary material, in §5 and §6 we generalize for ordered abelian groups and ordered domains the just-said result for ordered fields, and in §7 we employ these generalizations to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. We further show that an initial subdomain of No is discrete if and only if it is an initial subdomain of No's canonical integer part Oz of omnific integers. And, in §8, we extend results of [15] by showing that the theories of divisible ordered abelian groups and real-closed ordered fields are the sole theories of ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of No. Finally, in §9 we state a pair of open questions regarding s-hierarchical ordered algebraic systems that supplement another open question raised in §8.
Throughout the paper, the underlying set theory is assumed to be NBG and as such by class we mean set or proper class, the latter of which, in virtue of the axioms of global choice and foundation, always has the "cardinality" of the universe of sets. For additional information on formalizing the theory of surreal numbers in NBG, we refer the reader to [12].

Preliminaries I: Lexicographically ordered binary trees and Surreal Numbers
A tree A, < s is a partially ordered class such that for each x ∈ A, the class {y ∈ A : y < s x} of predecessors of x, written 'pr A (x)', is a set well ordered by < s . A maximal subclass of A well ordered by < s is called a branch of the tree. Two elements x and y of A are said to be incomparable if x = y, x < s y and y < s x. An initial subtree of A, < s is a subclass A ′ of A with the induced order such that for each and a root of A is a member of the zeroth level. If x, y ∈ A, then y is said to be an immediate successor of x if x < s y and ρ A (y) = ρ A (x) + 1; and if (x α ) α<β is a chain in A (i.e., a subclass of A totally ordered by < s ), then y is said to be an immediate successor of the chain if x α < s y for all α < β and ρ A (y) is the least ordinal greater than the tree-ranks of the members of the chain. The length of a chain (x α ) α<β in A is the ordinal β.
A tree A, < s is said to be binary if each member of A has at most two immediate successors and every chain in A of limit length has at most one immediate successor. If every member of A has two immediate successors and every chain in A of limit length (including the empty chain) has an immediate successor, then the binary tree is said to be full. Since a full binary tree has a level for each ordinal, the universe of a full binary tree is a proper class.
Following [15,Definition 1], a binary tree A, < s together with a total ordering < defined on A will be said to be lexicographically ordered if for all x, y ∈ A, x is incomparable with y if and only if x and y have a common predecessor lying between them (i.e. there is a z ∈ A such that z < s x, z < s y and either x < z < y or y < z < x). The appellation "lexicographically ordered" is motivated by the fact that: A, <, < s is a lexicographically ordered binary tree if and only if A, <, < s is isomorphic to an initial ordered subtree of the lexicographically ordered canonical full binary tree B, < lex(B) , < B , where B is the class of all sequences of −s and +s indexed over some ordinal, x < B y signifies that x is a proper initial subsequence of y, and (x α ) α<µ < lex(B) (y α ) α<σ if and only if x β = y β for all β < some δ, but x δ < y δ , it being understood that − < undefined < + [15, Notational Conventions. Let A, <, < s be a lexicographically ordered binary tree. If (L, R) is a pair of subclasses of A for which every member of L precedes every member of R, then we will write 'L < R'. Also, if x and y are members of A, then 'x < s y' will be read "x is simpler than y"; and if there is an x ∈ I = {y ∈ A : L < {y} < R} such that x < s y for all y ∈ I − {x}, then we will denote this simplest member of A lying between the members of L and the members of R by '{L | R}'. Finally, by 'L s(x) ' we mean {a ∈ A : a < s x and a < x} and by 'R s(x) ' we mean {a ∈ A : a < s x and x < a}.
The following three propositions collects together a number of properties of, or results about, lexicographically ordered binary trees that will be appealed to in subsequent portions of the paper.
Let No, <, < s be the lexicographically ordered binary tree of surreal numbers constructed in any of the manners found in the literature ( [14], [15], [16], [19], [10], [24]), including simply letting No, <, < s = B, < lex(B) , < B . 1 Central to the development of the s-hierarchical theory of surreal numbers is the following result where a lexicographically ordered binary tree A, <, < s is said to be complete [15,Definition 6], if whenever L and R are subsets of A for which L < R, there is an x ∈ A such that x = {L | R}.

An immediate consequence of Proposition 2 is
Proposition 3. Let A, <, < s be a lexicographically ordered binary tree. A, < s is full if and only if A, < is an η On -ordering (i.e. whenever L and R are subsets of A for which L < R, there is an x ∈ A such that L < x < R).

Preliminaries II: Conway names
Let D be the set of all surreal numbers having finite tree-rank, and 1 Conway's cut construction [9] and the related construction based on Cuesta Dutari cuts introduced in [11] (and adopted in [2] and [3]), do not include No's lexicographically ordered binary tree structure. However, as was noted in [14, page 257], they admit relational extensions to the ordered tree structure vis-á-vis the definition: for all x = (L, R), y ∈ No, x < s y if and only if L < {y} < R and y = x.
The following result regarding the structure of R is essentially due to Conway [9, pages 12, 23-25]. No's canonical class On of ordinals consists of the members of the "rightmost" branch of No, <, < s , i.e. the unique branch of No, <, < s whose members satisfy the condition: x < y if and only if x < s y. In those formulations where surreal numbers are pairs (L, R) of sets of previously defined surreal numbers ( [9], [19], [2]), the ordinals are the surreal numbers of the form (L, ∅); and in the formulation [20] where surreal numbers are signexpansions (see §2), the ordinals are the sequences (including the empty sequence) consisting solely of +s.
A striking s-hierarchical feature of No is that every surreal number can be assigned a canonical "proper name" (or normal form) that is a reflection of its characteristic s-hierarchical properties. These Conway names, as we call them, are expressed as formal sums of the form α<β ω yα .r α where β is an ordinal, (y α ) α<β is a strictly decreasing sequence of surreals, and (r α ) α<β is a sequence of nonzero real numbers, the Conway name of an ordinal being just its Cantor normal form [9, pages 31-33],[15, §3.1 and §5].
The surreal numbers having Conway names of the form ω y are called leaders since they denote the simplest positive members of the various Archimedean classes of No. More formally, they may be inductively defined by formula where n ranges over the positive integers, and a L and a R range over the elements of L s(y) and R s(y) , respectively. There are a number of significant relations between surreal numbers that are reflected in terms of relations between their respective Conway names. The following collection of such results, which are known from the literature, will be appealed to in the subsequent discussion. (ii) α<µ ω yα · r α < s α<β ω yα · r α whenever µ < s β; if β is a limit ordinal (where n and µ range over all positive integers and all ordinals less than β, respectively).
We shall also appeal to the following compilation of results regarding Conway names which, while new to the literature, essentially consists of corollaries of results from the first author's analysis of the surreal number tree [18] or improvements (based on that analysis) of a result of Gonshor [20, Lemma 5.8(a)].
moreover, in virtue of (i), ω y · r L < s ω y · r and ω y · r R < s ω y · r for all r L ∈ L s(r) and r R ∈ R s(r) . (iii) If r ∈ D − Z and L s(y) = ∅, then where n ranges over N; moreover, ω y · r L + ω y L · n < s ω y · r and ω y · r R − ω y L · n < s ω y · r, for all y L ∈ L s(y) , r L ∈ L s(r) and r R ∈ R s(r) . (iv) For all n ∈ N, 1 2 n ω y < s ω x if y ∈ R s(x) . Proof. (i) follows immediately from [18, Theorems 3.13 and 3.16], by considering a number c ∈ R − D such that b = c or b < s c. Necessarily, ω y · c is the immediate successor of a chain of limit length having a cofinal subchain of the form (ω y · c n ) n<ω where c is the immediate successor of (c n ) n<ω . As a ∈ (c n ) n<ω and either b ∈ (c n ) n<ω or b = c, it is the case that ω y · a ∈ (ω y · c n ) n<ω and either ω y · b ∈ (ω y · c n ) n<ω or ω y · b = ω y · c. In either case, ω y · a < s ω y · b. For (ii), if r ∈ R − D, the result can be proved from (i) by simply forming the specified cut in the just-said cofinal subchain; and if L s(y) = ∅ and r ∈ D − Z, the result follows from (i) and the second part of Gonshor's Lemma 5.8(a) from [20]. (iii) follows from the first part of Gonshor's Lemma 5.8(a) from [20] and repeated applications of [18,Theorem 3.18(ii)]. For (iv), suppose without loss of generality that x is the immediate left successor of y; for if it is not, the immediate left successor of y is in R s(x) and if we show that 1 2 n ω y is simpler than the immediate left successor of y for each n, then it follows that 1 2 n ω y is simpler than x as well. Consider the chain pr No (ω x ) of predecessors of ω x . Clearly, this is a chain of limit length, as ω x is the immediate successor of another surreal number if and only if x = 0, in which case R s(x) = ∅. By [18, Theorems 3.13 and 3.16], we may conclude that pr No (ω x ) contains a cofinal chain of the form (ω y · a n ) n<ω where a n = 1 2 n for each n. As this chain is contained in pr No (ω x ), 1 2 n ω y < s ω x for each n.
Let R((t Γ )) On be the ordered group (ordered domain; ordered field) of power series (definedá la Hahn [21]) consisting of all formal power series of the form α<β r α t yα where (y α ) α<β∈On is a possibly empty descending sequence of elements of an ordered class (ordered monoid; ordered abelian group) Γ and r α ∈ R − {0} for each α < β. R((t Γ )) On is a set (often simply written R((t Γ ))) if Γ is a set, and a proper class otherwise. An element For a truncation closed, cross sectional subgroup (subdomain; subfield) A of R((t Γ )) On , the set R y = {r ∈ R : rt y ∈ A} is an Archimedean ordered group (domain; field) which we will call the y-coefficient group (domain; field) of A.
Proposition 6 ( [15], [19]). There is an isomorphism of ordered groups from No onto R((t No )) On that sends each surreal number α<β ω yα · r α to α<β r α t yα . The isomorphism is in fact an isomorphism of ordered domains and, hence, of ordered fields.

Preliminaries III: s-hierarchical ordered structures
Following [15, Definition 2], A, +, <, < s , 0 is said to be an s-hierarchical ordered group if (i) A, +, <, 0 is an ordered abelian group; (ii) A, <, < s is a lexicographically ordered binary tree; and (iii) for all x, y ∈ A A, +, ·, <, < s , 0, 1 will be said to be an s-hierarchical ordered domain if (i) A, +, ·, <, 0, 1 is an ordered domain; (ii) A, +, <, < s , 0 is an shierarchical ordered group; and (iii) for all x, y ∈ A Moreover, A, +, ·, <, < s , 0 will be said to be an s-hierarchical ordered Kmodule if (i) K is an s-hierarchical ordered domain, (ii) A is an s-hierarchical ordered group, and (iii) A is an ordered K-module in which for all x ∈ K and all y ∈ A s-hierarchical ordered domains and modules are generalizations of the shierarchical ordered fields and vector spaces introduced in [15]. In virtue of Conway's field operations, No, +, ·, <, < s , 0, 1 is an s-hierarchical ordered domain. In fact, it is (up to isomorphism) the unique universal and unique maximal s-hierarchical ordered domain (in the sense of [15, page 1239]). Moreover, if K is an s-hierarchical ordered subdomain of No, then No is an s-hierarchical ordered K-module. Furthermore, extending the argument for s-hierarchical ordered subfields and subspaces of No from [15, page 1236], it is evident that Proposition 7. The s-hierarchical ordered subdomains and submodules of No coincide with the initial subdomains and the initial submodules of No, respectively.
Extending the notation employed in [15], if A is an ordered module and B ⊆ A, then by (B) A we mean the ordered submodule of A generated by B.
The next preparatory result is a modest generalization of [15,Theorem 6].
Proof. Except for replacing the references to "K-vector spaces" and "subspaces" with references to "K-modules" and "submodules", the proof is the same as the proof of [15, Theorem 6].

Initial subgroups and submodules of No
To fully characterize the initial subgroups of No, we must first characterize the initial subgroups of R.
It follows from the definition of R and Proposition 4 that the subgroups of R specified in the statement of the lemma are initial. Now suppose G is a nontrivial initial subgroup of R that does not contain D. Then there is a greatest m ∈ N such that z 2 m ∈ G for some z ∈ Z. Using closure under subtraction and the fact that we must always have 1 and, therefore Z in any nontrivial initial subgroup of No, it follows that 1 2 m ∈ G and, by closure under addition and subtraction, { z 2 m : z ∈ Z} ⊆ G. But since m is the greatest member of N for which z 2 m ∈ G for some z ∈ Z, the inclusion is not proper, thereby proving the lemma.
Our final preparatory result follows immediately from the definitions. Proposition 8. If G is a cross sectional, truncation closed subgroup of R((t Γ )) On and Z = µ<ν r µ t yµ ∈ G : ν is an infinite limit ordinal and r 0 = 1 , then {rt y : y ∈ Γ, r ∈ R y } ∪ Z constitutes a class of generators for G considered as a Z-module.
As is noted above, No is isomorphic to R((t No )) On . We now prove more generally Theorem 1. An ordered abelian group is isomorphic to an initial subgroup of No if and only if it is isomorphic to a truncation closed, cross sectional subgroup G of a power series group R((t Γ )) On , where (i) Γ is isomorphic to an initial ordered subclass of No, (ii) every y-coefficient group R y of G is an initial subgroup of R, and (iii) D ⊆ R y for all x, y ∈ Γ where y ∈ R s(x) .
Proof. Let A be an initial subgroup of No, Lead(A) be the class of leaders in A, and Γ = {y : ω y ∈ Lead(A)}. Following [15,Definition 14], a class B of surreal numbers is said to be approximation complete if α<σ ω yα · r α ∈ B whenever α<β ω yα · r α ∈ B and σ < β. Since A is initial, it follows from parts (i) and (ii) of Proposition 5 that A is approximation complete and Γ is initial. Moreover, since A is a group, for each y ∈ Γ the set R y = {r ∈ R : ω y · r ∈ A} is a subgroup of R. Furthermore, since A is initial, it follows from Lemma 1(i) that R y is itself initial. Now suppose x, y ∈ Γ where y ∈ R s(x) . In virtue of Lemma 1(iv), 1 2 n ω y < s ω x for each n ∈ N. Therefore, 1 2 n ∈ R y for each n ∈ N; and thus, since groups are closed under addition and subtraction, D ⊆ R y . Accordingly, by appealing to the restriction to A of the isomorphism specified in Proposition 6, it is evident that G = { α<β r α t yα ∈ R((t Γ )) On : α<β ω yα · r α ∈ A} is a truncation complete, cross sectional subgroup of R((t Γ )) On having the requisite properties listed in the statement of the theorem, thereby establishing the "only if" portion of the theorem. Aspects of the "if" portion of the proof borrow from the first author's proof of [15,Theorem 18]. However, whereas the latter proof concerns an ordered subfield F of R((t Γ )) On , which is treated as an ordered vector space over the Archimedean ordered field {r ∈ R : rt 0 ∈ F }, here we are concerned with an ordered subgroup G of R((t Γ )) On that we may only assume to be an ordered Z-module, which complicates the argument. To keep the argument largely self-contained, however, we repeat with modifications portions of the earlier proof.
Let G be a subgroup of R((t Γ )) On satisfying the conditions specified in the statement of the theorem and let A be the isomorphic copy of G in No that is the image of the restriction to G of the inverse of the mapping referred to in Proposition 6. That is, let A = { α<β ω yα · r α ∈ No : α<β r α t yα ∈ G}. To show that A is an initial subgroup of No, it suffices to show that A, < s |A is an initial subtree of No, < s . We do this by induction on Γ.
Let a 0 , . . . , a α , . . . (α < β) be a well-ordering of Γ such that ρ No (a µ ) ≤ ρ No (a ν ) whenever µ < ν < β. We consider A as an ordered Z-module. Let A α be the submodule of A containing 0 as well as all of the elements in A with exponents only from Γ α = {a δ : δ ≤ α}. We see that, considered as an ordered Z-module, A = α<β A α . Notice also that since a 0 = 0, A 0 = R 0 , which, by condition (ii), is an initial subgroup of No. Therefore, A 0 , < s |A 0 is an initial subtree of No. To complete the proof that A is an initial subtree of No, it remains to show that for all 0 < α < β, A α , < s |A α is an initial subtree of No, if µ<α A µ , < s | µ<α A µ is an initial subtree of No.
By appealing to Proposition 8 (and recalling that (X) A denotes the ordered submodule of A generated by X), it is easy to see that {ω aα · r α : r α ∈ R aα } ∪ Z α ∪ µ<α A µ is a class of generators for A α , and hence that, Thus to show that A α is an initial subtree of No, it suffices to show that B σ is an initial subtree of No for each σ < τ . Moreover, since B σ = δ<σ B δ , whenever b σ ∈ δ<σ B δ , henceforth we need only consider those b σ / ∈ δ<σ B δ . First, note that b 0 = ω aα . Moreover, since Γ is assumed to be initial, both L s(aα) and R s(aα) ⊆ {a δ : δ < α}. Let a L and a R be typical elements of L s(aα) and R s(aα) , respectively. It follows from Equation (1) where n ranges over the positive integers. As every element of µ<α A µ − {0} is Archimedean equivalent to a unique member of {ω a δ : δ < α} ⊆ µ<α A µ − {0}, there is be a unique partition (L α , R α ) of µ<α A µ where L α < R α , {0, nω a L } is cofinal with L α and { 1 2 n ω a R } is coinitial with R α . In virtue of the well ordering, each nω a L is in µ<α A µ as is each ω a R . Plainly then, {nω a L } ⊆ L α . Moreover, in virtue of the well ordering and the fact that condition (iii) in the statement of the theorem requires D ⊆ R a R for each a R , it follows that { 1 2 n ω a R } ⊆ R α . But then by Proposition 1, b 0 = {L α | R α }; and so, by Lemma 2, B 0 is an initial subtree of No.
If {ω aα · r α : r α ∈ R aα − Z} = ∅, we turn to Z α . Otherwise, we take the first remaining b σ ∈ {ω aα · r α : r α ∈ R aα − Z} for which b σ / ∈ δ<σ B δ and observe that, in virtue of the nature of our well ordering, b σ = ω aα · r α for some r α ∈ D−Z since every r α ∈ R−D has predecessors in D−Z. Moreover, either L s(aα) = ∅ or L s(aα) = ∅. If L s(aα) = ∅, then by the relevant portion of Lemma 1(ii), we have where ω aα · r L α , ω aα · r R α < s b σ ; and this together with the nature of our well ordering implies ω aα · r L α , ω aα · r R α ∈ δ<σ B δ . Furthermore, if L s(aα) = ∅, then by Lemma 1(iii), we have where ω y · r L + ω y L · n < s ω y · r and ω y · r R − ω y L · n < s ω y · r, for all y L ∈ L s(y) , r L ∈ L s(r) , r R ∈ R s(r) and n ∈ N; and, again, this together with the nature of our well ordering implies that all those options are contained in δ<σ B δ . Thus, in virtue of Proposition 1 (iii), in both cases there is a partition (L ′ σ , R ′ σ ) of δ<σ B δ such that b σ = {L ′ σ | R ′ σ }; and so, by Lemma 2, B σ is an initial subtree of No. This portion of the proof is repeated until each ω aα · r α for some r α ∈ D − Z has been dealt with.
Next, if {ω aα · r α : r α ∈ R aα − D} = ∅, we turn to Z α . If not, we take the first remaining b σ ∈ {ω aα · r α : r α ∈ R aα − D} for which b σ / ∈ δ<σ B δ and argue exactly as we did in the case where r α ∈ D − Z and L s(aα) = ∅ except we appeal to the relevant portion of Lemma 1(ii) concerned with members of R − D. This portion of the proof is repeated until each ω aα · r α for some r α ∈ R − D has been dealt with.
Finally, if Z α = ∅, we are finished; if not, let 0 < σ < τ and, as our induction hypothesis, suppose δ<σ B δ is an initial subtree of No. Since 0 < σ < τ , b σ has a Conway name of the form α< π ω yα .r α , where π is an infinite limit ordinal and r 0 = 1. Moreover, by part (ii) of Proposition 5, But since, by construction, Therefore, by virtue of the induction hypothesis and Lemma 2, B σ is an initial subtree of No. Thus, by induction, B σ is an initial subtree of No for each σ < τ , and so A α and, hence, A, < s |A are initial subtrees of No; thereby proving the theorem.
Remark. Using the axiom of choice or global choice (if the class is a proper class), it is a routine matter to prove that every ordered class is isomorphic to an initial ordered subclass of No. This might seem to suggest that in the statement of Theorem 1 one may omit the assumption that Γ is isomorphic to an initial ordered subclass of No. However, the presence of condition (iii) precludes the omission of that statement.

Densely and discretely ordered initial subgroups of No.
A nontrivial ordered group G, <, +, 0 is said to be discrete or discretely ordered if it contains a least positive member, and it is said to be dense or densely ordered if for all a, b ∈ G where a < b there is a c ∈ G such that a < c < b. A nontrivial ordered group G, <, +, 0 is dense if and only if it is not discrete.
The following proposition provides a simple means of distinguishing between nontrivial initial subgroups of No that are discrete and those that are dense. Proposition 9. An initial subgroup G of No is discrete if and only if there is a member of G of the form 1 2 n ω −α (where n ∈ N and α ∈ On) having no left immediate successor in G.
Proof. Note that elements of No of the form 1 2 n ω −α are precisely those having 0 as their sole left predecessor. We must show that for any g ∈ G, g is the least positive element of G if and only if L s(g) = {0} and g has no left immediate successor.
First, suppose there is a least positive g ∈ G for which L s(g) = {0}. If 0 / ∈ L s(g) , then g is not positive, a contradiction. If there is an a ∈ L s(g) where a = 0, then a is a positive element less than g, another contradiction.
Next, notice that any least positive g ∈ G must have no immediate left successor, since, if g has a left immediate successor, this successor must be a positive element less than g.
To show the other direction, suppose G is initial. Further suppose g has no left immediate successor, L s(g) = {0} and there is an a ∈ G where 0 < a < g. Since G is lexicographically ordered, it follows that for any x, y ∈ G where x < y, x is incomparable with y if and only if x and y have a common predecessor z such that x < z < y (see §2). Clearly, a must be incomparable with g, as the only surreal less than g and comparable with g is zero. Therefore, they must have a common predecessor z such that a < z < g, but by the assumption that L s(g) = {0}, z must equal 0 and a must be negative, which is impossible.

Initial subdomains of No
We now turn to the characterization of initial subdomains of No beginning with the initial subdomains of R. Proof. Let K be a subdomain of R. If K = Z, then by Proposition 4 K is initial. Now suppose K = Z. If K is initial, then Z ⊂ K and, by Proposition 4, K must contain some element of the form z + 1 2 where z ∈ Z. By subtracting z, we see that 1 2 ∈ K. But since the domain D is generated by 1 2 , D ⊆ K. Finally, if D ⊆ K, then K is initial since every every predecessor of a member of R − D is a member of D, and D is initial. The second part of the lemma is trivial.

Theorem 2. An ordered domain is isomorphic to an initial subdomain of
No if and only if it is isomorphic to a truncation closed, cross sectional subdomain K of a power series domain R((t Γ )) On , where Γ is isomorphic to an initial submonoid of No, every y-coefficient domain R y of K is an initial ordered subdomain of R, and D ⊆ R y for any x, y ∈ Γ where y ∈ R s(x) .
Proof. First note that the above conditions are precisely the conditions for initial groups, with the exception of the stipulation that Γ must be form a monoid. The latter condition is necessary since, if the image of the domain K is cross sectional, then for all x, y ∈ Γ, t x and t y are in K, so t x · t y = t x+y is in K and x + y is in Γ. Of course, the rest of the conditions are necessary for integral domains as well, as they are necessary for abelian groups. In order to show that the conditions are also sufficient, we may treat K as a Z-module and repeat the second part of the proof of Theorem 1.
6.1. Densely ordered initial subdomains of No. An ordered domain is said to be dense if its ordered additive group is dense.

Corollary 1. A densely ordered domain is isomorphic to an initial subdomain of
No if and only if it is isomorphic to a truncation closed, cross sectional subdomain K of a power series domain R((t Γ )) On where Γ is isomorphic to an initial submonoid of No and D is an initial subdomain of K.
Proof. Let K be a dense initial subdomain of No. In light of Theorem 2, to prove the corollary it suffices to show that D is an initial subdomain of K if and only if (i) every y-coefficient domain R y of K is an initial ordered subdomain of R, and (ii) D ⊆ R y for any x, y ∈ Γ where y ∈ R s(x) . Suppose D is an initial subdomain of K. Then D ⊆ R 0 . Moreover, since K is cross sectional, R 0 ⊆ R y for all y ∈ Γ, since t y ∈ K and r · t 0 ∈ K for all r ∈ R 0 , which implies r · t y ∈ K for all r ∈ R 0 . This implies (ii) is satisfied, which along with Lemma 4 implies (i) is satisfied as well. Now suppose (i) and (ii) are the case. If D is not an initial subdomain of K, then by (i) and Lemma 4, R 0 = Z. Moreover, since K is both dense and initial, the simplest member of No lying between 0 and 1, namely 1 2 , is in K. But this implies D is an initial subdomain of K, contrary to assumption.
The following result, which is a special case of Corollary 1, is the aforementioned result (see §1) categorizing the initial subfields of No established in [15]. Since the special case is about ordered fields, the ordered monoid Γ must be an ordered abelian group and the reference to D may be deleted since every ordered field is an extension of an isomorphic copy of D, the latter of which is initial in No.
Corollary 2. An ordered field is isomorphic to an initial subfield of No if and only if it is isomorphic to a truncation closed, cross sectional subfield of a power series field R((t Γ )) On where Γ is isomorphic to an initial subgroup of No.

Discretely ordered initial subdomains of
No. An ordered domain is said to be discrete if its additive group is discrete. Accordingly, an ordered domain is discrete if and only if it is not dense. The least positive member of an ordered domain is its multiplicative identity 1.
A discrete subdomain A of an ordered field B is said to be an integer part if every member of B is at most a distance 1 from a member of A. Conway introduced a canonical integer part Oz of No consisting of the surreal numbers of the form x = {x − 1 | x + 1}. These omnific integers, as Conway calls them, are precisely the surreal numbers having Conway names of the form α<β ω yα · r α where y α ≥ 0 for all α < β, and r α is an integer if y α = 0.
As we will now see, Oz is in fact (up to isomorphism) the unique discrete s-hierarchical ordered domain containing an initial isomorphic copy of every discrete s-hierarchical ordered domain.

Theorem 3. An initial subdomain of No is discrete if and only if it is an initial subdomain of Oz.
Proof. Suppose K is an initial subdomain of Oz. As Oz is an initial subdomain of No [18, p. 3: Note 2], K must be as well. To see that K is discrete, suppose on the contrary there is an element in K between 0 and 1. Since For the converse, suppose that there is a discrete initial subdomain K of No containing some element a / ∈ Oz. Let α<β ω yα · r α be the Conway name of a. Also let b = α<β ω yα · r ′ α where r ′ α = r α if y α > 0, r ′ α = 0 if y α < 0 and r ′ α is largest integer less than r α if y α = 0. Note that b ∈ Oz, so b = {b − 1 | b + 1}. Moreover, since b − 1 < a < b + 1, b < s a; and so, as K is initial and a ∈ K, b ∈ K. As K is a domain, a − b ∈ K; but 0 < a − b < 1, so 1 2 < s a − b and, therefore, 1 2 ∈ K, which contradicts the assumption that K is discrete.

Initial subgroups and subdomains that are convex
Among the important subgroups and subdomains of ordered groups and fields are those that are convex. Using Theorems 1 and 2, we now identify the convex subgroups of initial subgroups of No that are themselves initial as well as the convex subdomains (i.e. the valuation rings) of initial subfields of No that are likewise initial. As we shall see, unlike the convex subgroups and subdomains of ordinary ordered groups and ordered fields, the initial convex subgroups and subdomains of initial subgroups and subfields of No are always well ordered by inclusion.
Let A be a nontrivial initial subgroup of No and On(A) =: On ∩ A be its subtree of ordinals. Following [15,Definition 19], A is said to be α-Archimedean if α is the height of On(A). A nontrivial initial subgroup of No is α-Archimedean if and only if for each x ∈ A there is a β ∈ On(A) such that −β < x < β [15,Theorem 24].
Ordinals of the form ω φ are said to be additively indecomposable since they are precisely the ordinals λ such that µ+ν < λ for all ordinals µ, ν < λ, and ordinals of the form ω ω φ are said to be multiplicatively indecomposable since they are precisely the ordinals λ > 1 such that µν < λ for all ordinals µ, ν < λ, where the just-said sums and products of ordinals are the familiar Cantorian operations. Every nontrivial initial subgroup (resp. initial subdomain) is ω φ -Archimedean for some nonzero ordinal (resp. nonzero additively indecomposable ordinal) φ; moreover, A is Archimedean if and only if A is ω-Archimedean [15,Theorem 24].
Let A be an ω φ -Archimedean initial subgroup of No and for each nonzero ordinal τ ≤ φ, let (iv) For each leader ω y ∈ A there is a unique leader ω α ∈ A where α is an ordinal < φ such that ω y ≤ ω α and ω α ≤ s ω y . If ω y = ω α , then ω α is the least ordinal > ω y . Moverover, if ω x < s ω y , then ω x ≤ ω α .
Proof. (i) is trivial, (ii) follows from the additively indecomposable nature of ω τ , and (iii) follows from the definition of A[ω τ ] and the initial nature of A. For (iv), note that for each z > 0 in No, there is a unique ordinal α such that z ≤ α and α ≤ s z; moreover, if y = α, then α is the least ordinal > y. But by [15,Theorem 11] and [18, Proposition 3.6 (i)] for all z, y ∈ No, z < s y if and only if ω z < s ω y , which implies the first two parts of (iv). Finally, suppose ω z < s ω y . If z is an ordinal, then plainly ω z ≤ ω α ; and if z is not an ordinal, then its sign-expansion begins with ω α pluses followed by a minus, which implies it is < ω α , thereby completing the proof. Proof. First suppose K is a nontrivial initial convex subgroup of A. Since K is a nontrivial initial subgroup of A, K is ω τ -Archimedean for some nonzero But then ω τ < s x, which implies K is not initial, since ω τ / ∈ K; and so K = A[ω τ ]. Now suppose K = A[ω τ ] for some additively indecomposable infinite ordinal ω τ ≤ ω φ . If τ = φ, then K = A, which is a trivial initial convex subgroup of A. Now suppose 1 ≤ τ < φ. In virtue of the definition of A[ω τ ], K is a convex subclass of A. Moreover, by Proposition 10(ii) and the convex nature of K, every element of A that is Archimedean equivalent to some x ∈ K is likewise in K. Plainly then, x + y ∈ K whenever x, y ∈ K, since x + y is in the Archimedean class containing ω z , where z is the maximal member in the supports of x and the supports of y, which shows K is a group. To establish K is initial, first note that since A is cross sectional and closed under truncation, it follows from the fact that K is a convex subgroup of A (in which every element of A that is Archimedean equivalent to some x ∈ K is likewise in K) that K is also cross sectional and closed under truncation. In addition, since K is a convex subgroup of A and A is initial, the y-coefficient groups of K satisfy conditions (ii) and (iii) of Theorem 1. Therefore, in virtue of Theorem 1, to complete the proof it remains to show {y ∈ No : ω y ∈ K} is an initial subtree of No. For this purpose, suppose ω y ∈ K and further suppose ω x < s ω y . Since x < s y if and only if ω x < s ω y for all x, y ∈ No, it suffices to show ω x ∈ K. To this end, note that by Proposition 10(iii), there is an ordinal β ∈ K such that ω y < β < ω τ . Moreover, by Proposition 10(iv), there is an ordinal ω α such that either ω α = ω y or ω α is the least ordinal > ω y and ω x ≤ ω α . Plainly then, ω x ≤ ω α ≤ β. And so, since K is a convex subgroup of A, ω x ∈ K.
Theorem 5. Let A be an ω φ -Archimedean initial subfield of No. Then K is an initial convex subdomain of A if and only if K = A[ω τ ] for some multiplicatively indecomposable ordinal ω τ ≤ ω φ .
Proof. Since A is a field, it follows that for each ω y ∈ K, {r ∈ R : ω y .r ∈ K} = {r ∈ R : ω 0 .r ∈ K} and so, for each leader ω y ∈ K, D is a subdomain of {r ∈ R : ω y .r ∈ K}. Accordingly, in virtue of Theorem 4 and Theorem 2, to complete the proof it suffices to show that K (which is convex) is a subdomain of A if and only if K = A[ω τ ] for some multiplicatively indecomposable ordinal ω τ ≤ ω φ . If ω τ is not multiplicatively indecomposable, there are ordinals α, β ∈ K where αβ > ω τ and, hence, αβ / ∈ K, which shows K is not a domain. Now suppose ω τ is multiplicatively indecomposable. If τ = 1, K consists of the finitely bounded members of A, in which case K is obviously a subdomain of A. Moreover, if 1 < τ < φ, then {ω β : β < τ } is a cofinal subclass of K without a greatest member. Accordingly, if x, y ∈ K, |x| ≤ some ω β ∈ K and |y| ≤ some ω γ ∈ K, where β, γ are ordinals < τ , and so |xy| ≤ ω β+γ < ω τ . But then ω β+γ and, hence, |xy| are in K, which suffices to show K is a domain.
Remark. Since every real-closed ordered field is isomorphic to an initial subfield of No, it is natural to inquire if this is true for real-closed ordered domains more generally, the latter of which coincide with the convex subdomains of real-closed ordered fields [7]. However, since the convex subdomains of real-closed ordered fields are not in general well-ordered by inclusion, Theorem 5 implies this is not the case.

Optimality results and an open question
As we mentioned above, in [15] it was shown that every divisible ordered abelian group (real-closed ordered field) is isomorphic to an initial subgroup (initial subfield) of No. The following result shows that in a important sense these results are optimal.
Let T OAG and T DOAG be the theories of ordered abelian groups and divisible ordered abelian groups in the language {≤, +, 0} of ordered additive groups, and let T OF and T RCF be the theories of ordered fields and real-closed ordered fields in the language {≤, +, ·, 0, 1} of ordered fields. Proof. In light of the above-mentioned results on divisible ordered abelian groups and real-closed fields from [15], it remains to consider the cases where T = T DOAG and T = T RCF . Let T OAG ⊆ T ⊂ T DOAG and M be a countable model of T . There is an elementary chain M α (α < On) of models of T such that for each α, M α is an ω α+1 -saturated elementary extension of M of power 2 ωα [8, Lemma 5.1.4]. Since each M α is an η α+1 -ordering ([8, Page 369]; also see [4]), the union M ′ of the chain is a model of T that is an η On -ordering (see Proposition 3). However, since No is a lexicographically ordered full binary tree and no lexicographically ordered binary tree contains a proper initial ordered subtree that is isomorphic to itself [15, Lemmas 1 and 2], it follows from Propositions 2 and 3 that an initial ordered subtree of No is an η On -ordering if and only if it is No itself. But since No is divisible and M ′ is not, there is no isomorphism of M ′ onto an initial subgroup of No. Except for trivial modifications, the same argument applies to (ii), which suffices to prove the theorem.
The above proof of Theorem 6 makes critical use of class models. Using a classical result from the theory of saturated models [8, Lemma 5.1.5] together with a natural generalization of Theorem 4 of ( [15], [17]), a variation of the above proof shows that an analog of Theorem 6 that applies solely to structures whose universes are sets can be established in NBG supplemented with the existence of an inaccessible cardinal < On or an instance of GCH. This naturally suggests: Question 1. Can an analog of Theorem 6 that applies solely to ordered abelian groups and ordered fields whose universes are sets be established in NBG?
An ordered field K is said to be n-real-closed [6, page 327] if every polynomial of degree at most n admitting a root in a real closure of K admits a root in K. Boughattas [6] has shown that, for each positive integer n, there is a model of the theory of n-real-closed fields whose universe is a set that does not have an integer part. Therefore, since every initial subfield of No has a canonical integer part [15,Theorem 25], it follows that for each positive integer n, there is a model of the theory of n-real-closed fields whose universe is a set that is not isomorphic to an initial subfield of No. This, however, does not provide a positive answer to the field portion of Question 1 since there are theories of ordered fields that are not equivalent to the theory of n-real-closed fields for any n. One can also find theories of ordered abelian groups having models whose universe are sets that are not isomorphic to initial subgroups of No. For example, if we say that an ordered abelian group is n-divisible if every element is divisible by n, then for each prime p > 2 the p-divisible ordered abelian group generated by 1 is not isomorphic to an initial subgroup of No. However, we are not aware of a proof that applies to every theory of ordered abelian groups lacking full divisibility, which leaves the group portion of Question 1 open as well.

Further open questions
Unlike discrete initial subdomains of No, discrete initial subgroups of No need not be subgroups of Oz. A case in point is the subgroup of No consisting of all elements of the form d + (a/ω), where d ∈ D and a ∈ Z. On the other hand, this discrete initial subgroup of No is isomorphic to an initial subgroup of Oz, as is evident from the mapping f (d+(a/ω)) = ω.d+a for all d ∈ D and all a ∈ Z. This motivates Question 2. Is every discrete initial subgroup of No isomorphic to an initial subgroup of Oz? Question 3. What is a set of conditions that are individually necessary and collectively sufficient for an arbitrary ordered monoid to be isomorphic to an initial submonoid of No?