Number Systems with Simplicity Hierarchies: A ... · A GENERALIZATION OF CONWAY'S THEORY OF SURREAL...

THE JOURNAL OF SYMBOLIC LOGIC

Volume 66, Number 3, Sept. 2001

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY'S THEORY OF SURREAL

NUMBERS

PHILIP EHRLICH

Introduction. In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including -co, co/2, 1/co, wS and co - 7T to name only a few. Indeed, 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 "number" fields be individually definable in terms of sets of von Neumann- Bernays-Godel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], 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 system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean orderedfield, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10], [12], [13]. 1

However, in addition to its distinguished structure as an ordered field, No has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or simplicity hierarchy, as we have called it [15], depends upon No's (implicit) structure as a lexicographically ordered 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,

Received September 2, 1998; revised March 29, 2000. Portions of this paper were presented at the 1998 ASL Spring Meeting in Los Angeles, the 1998

ASL Summer Meeting in Prague, the 1999 Mal'tsev Meeting in Novosibirsk, and the University of Notre Dame Mathematical Logic Seminar. Research supported by the National Science Foundation (Scholars Award # SBR 9602154) and Ohio University. The author wishes to express his thanks to these institutions for their support and to Lou van den Dries and the referee for suggesting helpful ways for streamlining and improving the exposition.

1 For the purpose of this paper, an ordered field (Archimedean ordered field) A is said to be homogeneous universal if it is universal every ordered field (Archimedean ordered field) whose universe is a set or a proper class of NBG can be embedded in A and it is homogeneous every isomorphism between subfields of A whose universes are sets can be extended to an automorphism of A. Since model theorists frequently use the above italicized terms in more general senses, in the model-theoretic settings of [10], [12], [13] and [14] the terms absolutely homogeneous universal, absolutely universal, and absolutely homogeneous were respectively employed in their steads.

(?) 2001, Association for Symbolic Logic 0022-4812/01/6603-001 5/$3.80

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1232 PHILIP EHRLICH

it being understood that x is simpler than y just in case x is a predecessor of y in the tree.

In [15], the just-described simplicity hierarchy was brought to the fore2 and made part of an algebraico-tree-theoretic definition of No. In the pages that follow, we introduce a novel class of structures whose properties generalize those of No so construed and explore some of the relations that exist between No and this more general class of s-hierarchical ordered structures as we call them. In ? 1 we define a number of types of s-hierarchical ordered structures groups, fields, vector spaces- as well as a corresponding type of s-hierarchical mapping, identify No as a complete s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space), and show that there is one and only one s-hierarchical mapping of an s-hierarchical ordered structure into No (or any complete s-hierarchical ordered structure, more generally). These mappings are found to be embeddings of their respective kinds whose images are initial subtrees of No, and this together with the completeness of No enables us to characterize No, up to isomorphism, as the unique complete as well as the unique nonextensible and the unique universal, s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space). Following this, in ?2 and ?4 we turn our attention to uncovering the spectrum of s-hierarchical ordered structures. Given the nature of No alluded to above, this reduces to revealing the spectrum of initial substructures of No, i.e., the subgroups, subfields, subspaces of No (considered as an s-hierarchical ordered algebraic structure) that are initial subtrees of No. Included among our findings are the following two results that were originally stated as conjectures by the author at the AMS special session on Surreal Numbers in January of 1989.

I. Every divisible ordered abelian group is isomorphic to an initial subgroup of No.

II. Every real-closed orderedfield is isomorphic to an initial subfield of No.

In ?3, as part of the groundwork for the proof of II, we provide novel proofs that each surreal number x can be represented by a unique formal sum which may be treated as a canonicalproper name of x and the closely related fact that No considered as an ordered field is isomorphic to the formal power series field JR (No)On.

In ?5, we generalize and amplify Conway's theories of ordinals and omnific integers by showing that every nontrivial s-hierarchical ordered group (s-hierarchical ordered field) A contains a cofinal, canonical subsemigroup (subsemiring) On(A)

the ordinalpart of A which in turn is contained in a discrete, canonical subgroup

2More specifically, in [15], following a suggestion of Conway, the just-described simplicity hierarchy was brought to the fore and thereby freed from the ambiguity that befalls it in Conway's own treatment in [7]. The ambiguity arises because remarks made in [7] make it possible (if not more likely) to interpret "x is simpler than y" as x has an earlier birthday than y, rather than in the manner specified above as Conway had intended (Private Conversation: see [15, pp. 257-258: note 1]). For some purposes the ambiguity is of little consequence. For example, one may show that No has precisely one automorphism that preserves simplicity regardless of which one of the above two interpretations of the simpler than relation is adopted [4], [5], [15]. On the other hand, as the succeeding pages only begin to show, from the standpoint of exploring the internal structure of No, it the tree-theoretic interpretation that is the more revealing. For treatments of No in which "x is simpler than y" is interpreted as x has an earlier birthday than y, see, for example, [4], [5], and [6].

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES 1233

(subring) Oz(A) of A the omnific integer part of A-in which for each x E A there is a z E Oz(A) such that z < x < z + e where e which is the simplest positive element of A is the least positive element of Oz(A). When A is a substructure of No, e is the surreal number 1 and the members of On(A) and Oz(A) are called ordinals and omnific integers, respectively Finally, in ?6 we specify directions for further research.

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 Ax- iom of Global Choice, always has the "cardinality" of the class On of all ordinals. Moreover, since the usual definition of a sequence is not a legitimate conception in NBG when proper classes are involved, we follow the standard practice of un- derstanding by a "structure" whose universe A is a proper class and whose finitary relations R,, 0 < a < fi E On, on A are classes (which may be operations or distinguished elements treated as special relations) the class (A x {0}) U R where R = Uo<B<fl (Rc, x {a}). Tuples involving proper classes, more generally, are likewise understood.

? 1. s-Hierarchical ordered algebraic structures and s-hierarchical morphisms. 1.1. Preliminaries: Lexicographically ordered binary trees. A tree (A, <,) is a

partially ordered class such that for each x E A, the class prA (x) = (y E A: y <, x) of predecessors of x is a set well ordered by <,. Two elements x and y of A are said to be incomparable if x + y, x g5 y and y g5 x. An initial subtree of A is a subclass A' of A with the induced order such that for each x E A', prA/ (x) = prA (x). The tree- rank of x E A, written PA (x), is the ordinal corresponding to the well-ordered set (prA (x) , <s), and the a th level of A, henceforth LevA (a), is (x E A: PA (x) = a). A root of A is a member of LevA (0), and the height of A is On, if LevA (a) #4 0 for each a e On, and min { a e On: LevA (a) = 0} otherwise. If x, y E A, then y is said to be an immediate successor of x if x <, y and PA (Y) = PA (x) + 1; and if (xc,),c<p is a chain in A (i.e., a subclass of A totally ordered by <,), then y is said to be an immediate successor of the chain if x, <, y for all a < fi and PA (Y) = min{f E On: a > PA (xma) for all a < PI.

A tree is said to be binary if every element has at most two immediate successors and every chain of infinite limit length has at most one immediate successor. Using the Axiom of Global Choice (or simply the Axiom of Choice if A is a set) a tree may be shown to be binary if and only if it is isomorphic to a canonical binary tree,

i.e., an initial subtree of KB =U<0ni {- +} <B), where x <B y signifies that x

is a proper initial subsequence of y (cf. [9], p. 216). By an ordered tree (A, <, <,), we mean a tree (A, <s) whose universe is totally

ordered by <. As is well known, every canonical binary tree can be totally ordered (lexicographically) in accordance with the condition: (Xc,)cl<,i <le, (ycX)C,<, if and only if x, = y, for all a < some P < min {,u, v}, but xp < yp, it being understood that - < undefined < +. Moreover, as Theorem 1 below makes clear, one may provide representation independent characterizations of such an ordered structure using either the idea of a convex subclass of an ordered class A, i.e., a subclass I of A in which z E I whenever x, y E I, z E A and x < z < y, or

1234 PHILIP EHRLICH

DEFINITION 1. A lexicographically ordered binary tree is an ordered binary tree (A, <, <,) in which for all x, y E A 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.

NOTATIONAL AND TERMINOLOGICAL CONVENTIONS. If L and R are subclasses of an ordered class (A, <), then 'L < R' signifies that every member of L precedes every member of R. Also, if x and y are members of an ordered tree (A, <, <s), then x <, y will be read "x is simpler than y", and an element x of a nonempty subclass I of A will be said to be "the simplest member" of I if x <, y for all y E I - {x}. Finally, by 'Ls(x)' we mean {y E A: y < x and y <, x} and by 'R,(x)' we mean {y E A: x < y and y <, x}. Thus, intuitively, Ls(x) is the set of all members of A that are both "to the left of x" (i.e., smaller than x) and simpler than x, and R,(x) is the set of all members of A that are both "to the right of x" (i.e., greater than x) and simpler than x.

THEOREM 1. Let (A, <, <,) be an ordered tree. Then the following three conditions are equivalent:

(i) (A, <, <s) is a lexicographically ordered binary tree; (ii) (A, <, <s) is isomorphic to an initial subtree of (B, <lex, <B);

(iii) every nonempty convex subclass of A contains a simplest member, andfor all x, y E A, Ls(x) < {y} < Rs(x) whenever x <s y.

Moreover, the isomorphism in (ii) is unique. PROOF. Suppose (i), and for each x E A let g (x) = (gc (X))0y<PA(X) where for all < PA (x)

g + if

x0 < x

g~x)=l_ ifx0> X,

where xo is the predecessor of x of tree-rank a. It is easy to see that g is the unique isomorphism of trees from (A, <s) onto an initial subtree of (B, <B). Now let x, y e A where x < y. A simple argument using Definition 1 shows that either (a) Ls(x) U {x} C Ls(Y) and Rs(x) C Rs(Y), or (b) Ls(Y) C Ls(x) and RS(W) U {y} C

Rs(x) or (c) Rs(x) n Ls(Y) 7 0. But if (a), gc, (x) = go, (y) for all a < PA (x),

gPA(X) (y) = +, and gPA(X) (x) is undefined; and if (b), gc, (x) = go (y) for all a < PA (y), gPA(Y) (x) = -, and gPA(Y) (y) is undefined; from whence, in each case, it follows that g (x) <lex g (y). Finally, if (c), gc, (x) = gc (y) for all a < some ,6 < min{PA (x) , PA (y)}, gp (x) = - and gp (y) = +; and so, again, g (x) <lex g (y); thereby proving (i) implies (ii). Now suppose (ii), and let (B', <lex', <B')

be an initial subtree of (B, <lex, <B) that is isomorphic to (A, <, <s). Clearly then, to show (ii) implies (iii), it suffices to show (B', <lex', <B/) has the properties specified in (iii). Accordingly, suppose I is a nonempty convex subclass of B', x c I' = {a e I: LevA (a) is minimal}, and x 5B/ y for some y e I - {x}. Plainly, x is incomparable with y, whence in virtue of the ordering there is a z e A such that

3Indeed, an ordered binary tree (A, <, <,) is lexicographically ordered just in case for all x, y E A, x < y if and only if either (a) or (b) or (c) is the case. It is also not difficult to show that an ordered binary tree (A, <, <,) is lexicographically ordered just in case for all x, y E A, x < y if and only if precisely one of the following is the case: x E Ls(Y) or y E Rs(x, or Rsf(xl Ls(Y) $ 0. In [15, p. 249], the author presumed but failed to explicitly state the qualification "precisely one" in the just stated alternative characterization of an isomorphic copy of an initial ordered subtree of (B, <lex, <B).


either z E Rs(x) 0 Ls(y) or z E Rs(y) 0 Ls(x) depending upon whether x <lex' y or Y <lex' x, respectively. But then LeVA (z) < LevA (x) and (since x <lex Z <lex' Y or y <lex' Z <lex' x) z E I contrary to the assumption that x E I'; thereby proving x is the simplest member of I. Now suppose x, y E I where x <B' y. By the definitions of <B' and <lex', Ls(x) C Ls(y) <lex' {y} <ley' Rs(y) D Rs(x) and, hence,

Ls(x) < lex' {y} <lex' Rs(x); thereby establishing (ii) implies (iii). Finally suppose (iii) and let x, y E A where x < y. If x <s y (y <s x), then Ls(x) < {y} < Rs(x) (Ls(y) < {x} < Rs(y)), and so Rs0(x n Ls(y) 0. Moreover, if x g5 y and y g5 x, then I = {a: Ls(X) < {a} < Rs(y) } :A 0 insofar as x, y E I; and so, by (iii), there is a simplest member of I, say, z. But since x is incomparable with y, it follows that z <s x and z <s y, and hence, by the definition of I, that z E Rs(x0 n Ls(y); thereby showing (iii) implies (i). -

NOTATIONAL AND TERMINOLOGICAL CONVENTIONS. Let (A, <, <s) be a lexicographically ordered binary tree. If x is the simplest member of A lying between two subsets L and R of A where L < R and {y E A: L < {y} < R} :A 0, then we will denote x by '{L I R}A, or, simply, by '{L I R}' when no ambiguity will arise. Moreover, applying Conway's conventions [7, p. 4] to our definition of '{L I R}', if x = {L I R} we write 'xL, for the typical member of L and 'xR, for the typical member of R; for x itself we write '{xL xR }'; and x = {a, b, c,... d, e, f, ... } means that x = {L I R} where a, b, c,... are the typical members of L and d, e, f,... are the typical members of R.

Central to the remainder of the paper is the fact that for each element x of a lexicographically ordered binary tree (A, <, <s), there are subsets L and R of A where L < R such that x = {L I R}. Indeed, we will have occasion to employ a variety of distinct such representations of x including the canonical representation, x = {Ls(x) I Rs(x) }. The justification of all such representations is provided by the first part of the following simple consequence of Theorem 1.

THEOREM 2. Let (A, <, <s) be a lexicographically ordered binary tree, x E A and (L, R) be a pair of subsets of A for which L < R. Then x = {L I R} if and only if L < {x} < R and {y E A: L < {y} < R} C {y E A: Ls(x) < {y} < Rs(x)}. Moreover, x <s y if and only ifLs(x) < {y} < Rs()andy : x.

Also of importance to portions of the subsequent discussion is the following consequence of last two parts of Theorem 2.

THEOREM 3. Let (A, <, <s) be a lexicographically ordered binary tree. Then y is an immediate successor of x in A if and only if either y = {Ls(x) U {x} I Rs(x) } or

y {Ls(x) {x} U Rs(x) }. Also, if (x,)c<p is a chain in A of infinite limit length,

then y = {W<p Ls(xf) lUcx<e Rs() } if and only if y is the immediate successor of the chain.

1.2. s-Hierarchical ordered algebraic systems: definitions, explanations and preliminary results. In Definitions 2-4 below, xL, XR, yL and yR are understood to designate typical members of arbitrary sets Lx, Rx, Ly, and Ry, respectively, for which x = {Lx I Rx} and y = {Ly Ry}.

1236 PHILIP EHRLICH

DEFINITION 2. (A, +, <, <0, 0) will be said to be an s-hierarchical ordered group if (i) (A, +, <, 0) is an ordered abelian group; (ii) (A, <, <,) is a lexicographically ordered binary tree; and (iii) for all x, y E A

X y XL + y, X + yL XR+ Y X + yR}.

DEFINITION 3. (A, +, , <, <,, 0, 1) will be said to be an s-hierarchical orderedfield if (i) (A, +, ., <, 0, 1) is an ordered field; (ii) (A, ?, <s, 0) is an s-hierarchical ordered group; and (iii) for all x, y E A

xy {xLy+xyL XLyL xRy+xyR x yR

x y + xyR -x LyR, xRy + XyL _ xRyL}.

DEFINITION 4. (A, ?,, <, <s) will be said to be an s-hierarchical ordered vector space over K if (i) K is an s-hierarchical ordered field, (ii) A is an s-hierarchical ordered group, and (iii) A is an ordered K-vector space in which for all x E K and ally e A

xy {xLy + xyL XLyL xRy + xyR -x RyR

x y + xy -x LyR, xRy + XyL _ xRyL}.

THE CONSTRAINTS ON SUMS AND PRODUCTS IN DEFINITIONS 2-4 are tree-theoretic adaptations of those employed by Conway [7, p. 5]. Their underlying motivation, which we will later have reason to invoke, may be encapsulated thus. Since x = {xL xR}A and y = {yL I yR}IA we have (*) xL < x < xR and yL < y < yR for all xL, XR yL and yR which together with the ordered group structure of A implies (**) {XL +y,X +yL} < {X +y} < {XR +y, X + yR}. Thus, since x +y must be a member of A lying between the two sets of members of A specified in (**), clause (iii) of Definition 2 requires that x + y be the simplest such member. Similarly, if A is an ordered field, it follows from (*) that (x - xL) (y _ yL) (XR _ X) (yR _ y)

(X - XL) (yR - y) and (xR - x) (y _ yL) are positive for all XL, XR, yL and

yR, which implies: (***) {xLy + xyL xLyL x Ry + xyR _ xRyR} < {xy} <

{xLy+xyR xLyR, xRy+ xyL_ xRyL}. Thus, since xy must lie between the two sets of members of A specified in (***), clause (iii) of Definition 3 requires that xy be the simplest member of A lying between those sets. Clause (iii) of Definition 4 is similarly motivated.

DEFINITION 5. A subgroup (subfield; subspace) B of an s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space) A will be said to be initial if B is an initial subtree of A.

It is evident from the explanation of the s-hierarchical constraints on sums and products given above that every initial subgroup (initial subfield; initial subspace) of an s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space) is itself s-hierarchical.

The three elementary consequences of Definitions 2 and 3 collected below will be employed in the subsequent discussion.


PROPOSITION 1. (i) 0 is the simplest element of (as well as the unique root in) an s-hierarchical ordered group A, i.e., 0 ={0 o 0}A; (ii) for each element x of an s-

hierarchical ordered group A, -x = {-xR I XL}A; (iii) 1 is the simplest positive

element of an s-hierarchical orderedfield A, i.e., 1 ={0 I}A.

1.3. Complete s-hierarchical ordered algebraic structures. A binary tree will be said to be full if every element has two immediate successors and every chain of infinite limit length < On has an immediate successor. As Theorem 4 below makes clear, in the case of lexicographically ordered binary trees the concept of a full binary tree is intimately related to the following conception.

DEFINITION 6. A lexicographically ordered binary tree (A, <, <,) will be said to be complete if whenever L and R are subsets of A for which L < R there is a y E A such that y = {L I R}. An s-hierarchical ordered structure will be said to be complete if it is complete as a lexicographically ordered binary tree.

It follows immediately from Definition 6 that no complete lexicographically ordered binary tree can be a set. More revealing, however, is

THEOREM 4. For a lexicographically ordered binary tree (A, <, <,) the following are equivalent:

(i) (A, <s) is full; (ii) (A, <, <s) is complete; (iii) the intersection of every nested sequence I, (0 < a < fi E On) of nonempty

convex subclasses of (A, <, <s) is nonempty (and hence, by Theorem 1, contains a simplest member).

PROOF. If (i), then (A, <, <s) is isomorphic to (B, <lex, <B), which along with Theorem 2.1 of [17] and parts (i) and (iii) of Theorem 1 implies (ii). And, if (ii) and I, (O < a < cl E On) is a nested sequence of nonempty convex subclasses of A, then I= nr,<, I, is itself a convex subclass of A. But if , = a + 1 for some a, then I = I,J : 0; and if f is an infinite limit ordinal and x, is the simplest member of I, for each a < fi, then I' -{a E A: Ua<p Ls)< <{a} < U<p Rs(x,) } C I; and so, by (ii), A, I' and, hence, I are nonempty, which proves (iii). Finally, if (iii), then for each x E A there are y, z E A where y = {Ls(x) U {x} I Rs(x) } and z {Ls(x) I{x} U Rs(x) }, and for each chain (x,)<, of infinite limit length in A, there is a w E A such that w { Ua<p Ls(X) U<p Rs(x) }. But, by Theorem 3, y and z are the immediate successors of x in A, and w is the immediate successor of the said chain in A, which implies (i).

Conway proved enough in [7] to show that No (together with its tree structure) is a complete s-hierarchical ordered field as well as a complete s-hierarchical ordered vector space over the unique initial subtree of No that is an isomorphic copy of the ordered field of reals (see Theorem 8 below); and alternative constructions of the ordered field No were provided by Gonshor [17], Ehrlich [11], [15] and Alling and Ehrlich [4], [6]. The following variation on results due to Conway [7, p. 65] and Gonshor [17] defines what, henceforth, we shall mean by No.

PROPOSITION 2. (No, <, <s, +,-, -) is a complete s-hierarchical orderedfield where (No, <, <s) = (B, <,ex, <B) and +, -, and are inductively defined as follows where

1238 PHILIP EHRLICH

XL, XR, YL, and yRrange over the members of Ls(x), Rs(x), Ls(Y), and Rs(Y), respectively.

DEFINITION OF X + Y.

x + y {X +LYy,X+ YL XR+ yX+yR}.

DEFINITION OF-X.4

-X = {xR I -X L}

DEFINITION OF Xy.

xy {xLy + xyLXLyLxRy + xyR R y R

x y + xy -x YRXY + xyL _xRyL}.

1.4. s-Hierarchical embeddings and complete s-hierarchical ordered algebraic structures. The present subsection culminates in a theorem that indicates that (No, <, <s, +,-, ) bears much the same relation to s-hierarchical ordered algebraic systems that the ordered field of reals bears to Archimedean ordered algebraic systems. Central to its formulation and proof are the following definition and subsequent two lemmas.

DEFINITION 7. A mapping f: A -> A' between two lexicographically ordered binary trees A and A' will be said to be s-hierarchical if for all x E A, f (x) =

{f (L) I f (R)}A whenever x = {L I RIA. If, in addition, A and A' are s- hierarchical structures and the mapping is also an embedding of ordered groups (ordered fields; ordered vector spaces) it will be said to be an s-hierarchical embedding.

Lemma 2 below brings to the fore just how distinguished s-hierarchical mappings are. To prepare the way for our proof of this result we will first relate s-hierarchical mappings to more classical ones.

LEMMA 1. Let (A, <, <A) and (A', <', <A') be lexicographically ordered binary trees and f: A -> A' be a mapping. The mapping f is s-hierarchical if and only if for all x, y E A, (i) f (x) <' f (y) whenever x < y, (ii) f (x) <A' f (y) whenever X <A y, and (iii) PA' (f (x)) = PA (X).

PROOF. Let f be s-hierarchical and x, y E A. If x < y, then (by Theorem 2)

x = {Ls(x) I Rs(x) U {y}} ; and so, f (x) = {f (Ls(x)) f f (Rs(X) U {y})} A',

which implies f (x) <' f (y). Moreover, since (by appealing to Theorems 2

and 3) it is easy to see that f ({0 1 0}A) = {f (0) I f (0)}A = {0 I 0}A and that f sends immediate successors to immediate successors and immediate successors of infinite chains of limit length to immediate successors of infinite chains of limit length, it follows that f (x) <A' f (y) if x <A y, and that PA (X) =

4The definition of -x given above (i.e., Conway's definition) is not actually required for the definition of No; following Gonshor [17], one can always show that the ordered semigroup that arises from + and < is a group by showing that -x is the sequence that arises from the sequence x by replacing the -s and +s in the latter sequence with +s and -s, respectively Moreover, having done so, one can always obtain Conway's formula via part (ii) of Proposition 1. On the other hand, given the elegance of Conway's definition and the fact that subtraction is required for the definition of multiplication, we have chosen to follow Conway's lead.


PA' (f (x)). Conversely, if f is a mapping of the said kind and x E A, then f (Ls(x)) <' {f(x)} <' f (Rs(x)) and prAl (f (x)) = f (Ls(x)) U f (Rs(x)), which

implies f (x) {f(Ls(x)) f (Rs(x)}A , which (by Theorem 2) suffices to prove the lemma. -

LEMMA 2. (i) Every s-hierarchical mapping between s-hierarchical ordered groups (s-hierarchical ordered fields; s-hierarchical ordered vector spaces over K) is an s- hierarchical embedding; (ii) iff: A -, A' andg: A -, A' are s-hierarchical mappings, then f = g and f (A) is an initial subtree of A'.

PROOF. Suppose f: A -, A' is an s-hierarchical mapping. Since both (ii) and the assertion that f is order preserving are immediate consequences of Lemma 1, to complete the proof it suffices to show by induction that f is an embedding of groups, fields, and vector spaces, respectively. The induction hypothesis for groups (fields; vector spaces) is that f has the desired property for all sums (products; products) at least one of whose summands (multiplicands; multiplicands) is simpler than x or simpler than y.' The proof for fields uses the result for groups and the proof for vector spaces mimics the proof for fields. -

DEFINITION 8. An s-hierarchical ordered group (s-hierarchical ordered field; s- hierarchical ordered vector space) A will be said to be universal if for each s- hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space) B there is an s-hierarchical embedding f: B -, A.

DEFINITION 9. An s-hierarchical ordered group (s-hierarchical ordered field; s- hierarchical ordered vector space) A will be said to be maximal or nonextensible if there is no s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space) that properly contains A as an initial subgroup (initial subfield; initial subspace).

THEOREM 5. Let A be an s-hierarchical ordered group (s-hierarchical orderedfield; s-hierarchical ordered vector space). Then A is complete if and only if A is universal if and only if A is maximal if and only if A is isomorphic to No.

PROOF. Let A and A' be s-hierarchical structures of the same type. In virtue of Theorem 2, to establish the existence of an s-hierarchical mapping f: A' -, A

it suffices to show f (x) = {f (Ls(x) I f (Rs(x) }A whenever x E A'. If A is complete, then (by virtue of the first part of Proposition 1 and Theorem 3) such a mapping may be defined by recursion as follows. Suppose f (y) has been de-

fined for all y E Ls(x) U Rs(x). If x = {0 1 0}A, let f (x) {f (0) I f (0)} =

{0 0}A; and if x is an immediate successor of some member of A', say, x', let

f (x) {f (Ls(x,) U { f (x')} f (Rs(X,)) }A when x = {Ls(xf) U {x'} I Rs(X,) }A

andf (x) {f (Ls(x)) I {f (x')} Uf (Rs(x,))}Awhenx {L(x) I {x'} U Rs(x)} and when x is the immediate successor of a chain (xc,),<p of infinite limit length where xc, is the member of prA (x) of tree-rank a, then x { Ua<f Ls(x)

The ordr p r v in which case of let t (x) tU<p th (Lse(xtees of A i es the The order preserving nature of f together with the completeness of A implies the

5For further discussions of the induction principles, see [7, p. 5] and especially [15, Appendix II].

1240 PHILIP EHRLICH

existence of f (x) so defined and thereby, in virtue of Lemma 2, the universal nature of A if A is an s-hierarchical ordered group or an s-hierarchical ordered field. For the case where A is an s-hierarchical ordered vector space, the universality of A is established by appealing to Lemma 2 in conjunction with the following result whose simple proof is left to the reader:

PROPOSITION. Let K be an s-hierarchical orderedfield, f be the unique s-hierarchical mappingfrom K to No, and No be the multiplication in the s-hierarchical orderedfield No. (No, +, <, <s, 0) together with the scalar product defined by the condition x y = f (x) Noy for x c K and y c No is a complete s-hierarchical ordered vector space over K.

On the other hand, if A is universal, there is an s-hierarchical embedding f: A' > A. But if A is a proper initial substructure of A', there are at least two distinct s- hierarchical embeddings from A into A, namely f I A and the identity map. But, by Lemma 2, this is impossible; and so A is maximal. Moreover, if we suppose A is maximal, it follows from the completeness of No (Proposition 2) and the already established fact that completeness implies universality that there is an s-hierarchical embedding f: A -* No. But since A is maximal, an argument like the one employed to establish that universality implies maximality shows that f is a surjection and, hence, that A is isomorphic to No and is, thus, complete. H

It is worth emphasizing that the s-hierarchical nature of A is crucial to the equiva- lences stated in Theorem 5. In connection with this, it is of interest to draw attention to

PROPOSITION 3. There is an orderedfield A D No that is not isomorphic to No.

PROOF. The idea of a field extension of a field whose universe is a proper class is in fact formalizable in NBG (cf. [16]). Accordingly, since No is real-closed [7, pp. 40-42] and no simple transcendental extension of a real-closed ordered field is itself real-closed, to obtain an ordered field of the desired kind we need only form the ordered field No(a) (of power On) where a is an indeterminate such that No < {a}. H

It is perhaps also worth noting that Proposition 3 takes on added significance since Dales and Woodin [8, p. 58] apparently misled by Conway's unfortunate choice of the phrase "universally embedding" to denote an embedding property of No called "universally extending" by the present author ([1 1]; [12]) have mistak- enly asserted that No is up to isomorphim the unique universal ordered field. (See note 2 below).

Thus far, the only s-hierarchical ordered algebraic structures whose existence we are assured of are complete s-hierarchical ordered structures. However, as we mentioned in the Introduction, there is a wealth of s-hierarchical ordered algebraic structures that fall short of completeness. In ?2 and ?4 we will begin the process of shedding light on the spectrum of such structures. As is evident from Lemma 2, Theorem 5 and the remark immediately following Definition 5, this reduces to revealing the spectrum of subgroups, subfields, and subspaces of No that are initial subtrees of No.


?2. The spectrum of s-hierarchical ordered vector spaces. 2.1. s-Hierarchical ordered vector spaces. There are a great many ordered vector

spaces that are not isomorphic to initial subspaces of No. The present subsection culminates with a result specifying necessary and sufficient conditions for those which are. To prepare the way for our proof of the said theorem, we first establish two preliminary results, the proof of the first of which makes use of the following

ELEMENTARY OBSERVATION. If A is an initial subtree of a lexicographically ordered binary tree A' and x ={L R}A', where (L, R) is a partition of A, then A U {x} is an initial subtree of A'.

PROOF. If a C prA (x), then Ls(a) C Ls(x) < {x} < Rs(x) D Rs(a). But x -

{L I R}A; and so, by Theorem 2, there is a z c L U R such that z > each x' c LS W and z < each x' c Rs(x), which with Theorem 2 implies a <s z and, hence, a c A (since L U R = A is an initial subtree of A'). -1

NOTATIONAL CONVENTION. If B is an ordered K-vector space and A is a subclass of B, then by (A)B we mean the ordered subspace of B generated by A. As is well known, if A is a subspace of B and b c B - A, then (A U {b})B

{rb + a: r C K, a c A}.

THEOREM 6. If A' is an s-hierarchical ordered vector space over K, A is an initial subspace of A' and b -{L I R}A where (L, R) is a partition of A for which L < R, then (A U {b})A, is an initial subspace of A'.

PROOF. Suppose K has height i, A has height a, and further let K: {r C K: PA (r) =i}, A, = {x c A: PA (X) v}, and ({ x a, <*) be the Cartesian product of i and a well-ordered by the condition: (/3', v') <* (, v) if and only if /3' < /3 or /3' = /3 and v' < v. Finally, for each (/3, v) c i x a, let A(#,) {rb + a: r C K and a c Av}, A<, ) U(H, )<(A) A(T1) and A? A(,) U A< ,u) < (g, v) (fan ) (fly), Since (A U {b})A, {= rb + a: r C K,a C A}, we then have (A U {b})A,

U(p v)etxu Av). Plainly then, to complete the proof, it suffices to prove: (a) for all (/3, v) c i x a, A< is an initial subtree of A' and for each x c A(#,v) there is a

partition (L', R') of A< such thatx {L' R'}A' . First, note that A LevA (v)

for each v < a. Clearly then, since A is an initial subtree of A', A< is an initial subtree of A' whenever (/3, v) <* (1, 0). Further note, A< A. Accordingly, to

(1,0)

establish the first part of (a) reduces to proving: if A< is an initial subtree of A'

where (1, 0) <* (/3, v), then A?<- is an initial subtree of A'. But, by virtue of the

"Elementary Observation", this-and indeed, all of (a) reduces to proving: if A< is an initial subtree of A' where (1, 0) <* (, v) and x c A(f, v), then x - {L' I R'} where (L', R') is the unique partition of A< for which L' < {xf} < R'. It is to this end that we now turn.

Since x c A(#,v) and (1, 0) <* (/3, v), x = rb + a for some r C Ra where fi> 0 and some a c Av. Moreover, by applying the s-hierarchical condition on sums to a and the result of applying the s-hierarchical condition on products to r and b, we

1242 PHILIP EHRLICH

obtain x = rb + a {L(x) I R(x) } where

L(x) = rLb + (rbL -r LbL + a) rb + a , rR b + (rbR -r RbR + a) }, R(x) {rLb + (rbR - rLbR + a) rb + aR, rRb + (rbR -r RbL + a) },

and rL, rR, bL, bR, aL and aR are typical members of Ls(I.), Rs(I.), Ls(b), Rs(b), Ls(a) and Rs(a), respectively But since A is a vector space over K, (rbL - rLbL + a), (rbR - rRbR + a), (rbR - rLbR + a) and (rbR - rRbL + a) are all in A, since r, rL, rR c K and a, bL, bR C A; and consequently, rLb + (rbL -rLbL + a), r b+ (rbRrRbR + a), rL b + (rbR -rLbR + a) and rRb + (rbR rRbL a) are all in A< since each such sum is in one or another A(YE) where y < f,. Moreover,

sums of the forms rb + a L and rb + a R are in A< since each such sum is in one or another A(p,,) where a < v; and as such L(x) U R(x) C A<,,. But then L(x) C L'

and R(x) C R'; and hence, since x {L(x) I R(x) } and L' < {x} < R', we have x = {L' R'. -1

Our second preparatory result generalizes a familiar embedding theorem for ordered vector spaces over the rationals.

PROPOSITION 4. Let A andB be ordered K-vector spaces, A' be an ordered subspace of A, andf: A' - B be an embedding of ordered K-vector spaces. Then, ifa C A-A' andh: A' U {a} l- B is an order injection that extends f, there is a unique embedding g: (A' U {a})A -* B of ordered K-vector spaces that extends h.

PROOF. Replace the ordered field of rationals by K in Priess-Crampe's proof [24, pp. 163-164] for ordered vector spaces over the rationals. -]

THEOREM 7. An ordered K-vector space A is isomorphic to an initial subspace of No if and only if K is isomorphic to an initial subfield of No.

PROOF. If an ordered K-vector space A is isomorphic to an initial subspace of No, then K admits a relational extension to an s-hierarchical ordered field. And so, by Lemma 2 and Theorem 5, K is isomorphic to an initial subfield of No. Now suppose A is an ordered K-vector space having basis B and K is isomorphic to an initial subfield of No. Then No is itself an ordered K-vector space, and to obtain an initial subspace of No that is isomorphic to A one need only form the union f: A -* No of a continuous chain f,1: A,, -* No of embeddings of ordered K- vector spaces, where fo (Ao = {O}) = {O} and f,+?I: A,,+I = (A,, U {x,,})A -3 No is the unique embedding extending the order injection h,1: A,, U {x,, } -* No that extends f,, where xu is the first remaining element in some specified well-ordering of B and h,, (x,) = { fu, (L,,) I f, (R,,) } where (L,,, Ru) is the unique partition of A,, for which L,, < {x,,u} < R,. Since f (A) is clearly isomorphic to A, it remains to show that f (A) is an initial subtree of NTo. But since fo (Ao) is an initial subtree of No and since the union of a continuous chain of initial subtrees of No is itself an initial subtree of No, to complete the proof we need only appeal to Theorem 6 in conjunction with the definitions of the h,, s and f?+ls. H

2.2. s-Hierarchical ordered vector spaces over archimedean ordered fields. To un- cover the spectrum of s-hierarchical ordered vector spaces over subfields of the reals we must first identify the s-hierarchical Archimedean ordered fields. For this purpose we require the following definition in which by a Dedekind gap of a densely


ordered class A we mean a pair of nonempty subclasses X and Y of A where X has no greatest member, Y has no least member, X U Y = A, and X < Y.

DEFINITION 10. Let R = D U {{X I Y}: (X, Y) is a Dedekind gap in ED} where D = {x c No: PNo (x) < co}; as such, R is the set of all members of Uf<0, a {-, +} that end neither in an infinite sequence of -s nor in an infinite sequence of +s.

Except for inessential changes, the following result concerning R is due to Conway [7, pp. 23-25].

PROPOSITION 5. R (with ?, -, and < defined a la No) is isomorphic to the ordered field of reals defined in any of the more familiar ways, ED being No's ring of dyadic rationals (i.e., rationals of the form m/2' where m and n are integers); n = {0. . . , n - I1 } {= n - 1 I n + 1 }for each positive integer n, and the remainder of the positive dyadics are the arithmetic means of their left and right predecessors of greatest tree-rank; e.g., 2 {0 1}; furthermore, for each r c R, r {r - r + 2 } where n ranges over the positive integers.

The first portion of Proposition 5 being its requisite justification, henceforth we shall adopt the following

CONVENTION. R and its suitably defined subsets Q and Z shall be referred to as the sets of real numbers, rational numbers, and integers, respectively; and R+, Q+, and E+ shall be their corresponding sets of positive elements.

Being an ordered field, No contains exactly one subring of dyadic rationals. On the other hand, it is easy to see that No contains an entire proper class of distinct isomorphic copies of the ordered field of reals. What distinguishes R from the rest is captured by the first part of

THEOREM 8. R is the unique isomorphic copy of the orderedfield of reals that is an initial subfield of No. In fact, every Archimedean orderedfield is isomorphic to exactly one initial subfield of No, the latter being an initial subfield of R.

PROOF. Let R' be an initial Dedekind complete subfield of No. Then ED c R' and for each Dedekind cut (X, Y) of ED where (X, Y) is a gap there is a unique z C R' such that X < {z} < Y. Since every member of R' - ED has tree-rank co and R - ED is the set of all members of No of tree-rank co that fill one or another Dedekind gap in ID, R' = R. Now suppose F is the unique subfield of R that is isomorphic to a given Archimedean ordered field. Then ID C F C R; and, since R and ID are initial subtrees of No and every predecessor of a member of R is a member of ID, the second part of the result follows. H

In virtue of Theorems 7 and 8 and the familiar fact that every divisible ordered abelian group may be regarded as an ordered vector space over the rational, we have

THEOREM 9. Every ordered vector space over an Archimedean orderedfield is isomorphic to an initial subspace of No; and, as such, every divisible ordered abelian group is isomorphic to an initial subgroup of No.

?3. The s-hierarchical ordered field No. As we have already mentioned, an ordered field admits a relational extension to an s-hierarchical ordered field if and only if it is isomorphic to an initial subfield of No. In ?4 below we will shed further light

1244 PHILIP EHRLICH

on these structures by identifying them with isomorphic copies of particular formal power series fields. One of the central components in our proof of the identification is the relation that exists between the formal power series fields in question and the complete s-hierarchical ordered field (No, +H, *H <H, <5), the latter of which together with its relation to (No, +, , <, <,) (Theorem 16) is the principal focus of the present section. In (No, +H, *H <H, <s), which (by Theorem 16) is extension- ally indistinguishable from (No, +, , <, <s), the operations +H and *H are defined on, and the relation <H is defined between, members of No denoted by their respective Conway names (Definition 13), each of the latter of which, in virtue of its particular form as a formal sum, is assigned to denote the unique surreal number having a canonical property that emerges from the structure of No considered as an s-hierarchical ordered vector space over R. That there is the appropriate correspon- dence between the formal expressions in question (called normalforms by Conway) and the members of No picked out by the relevant kind of definite descriptions is assured by Theorem 14 which, like Theorem 16, is inspired by a result due to Conway [7, p. 33]. Proofs of nontree-theoretic versions of the latter two results have already been given by Alling [3] and Gonshor [17], but their treatments are quite different, longer, and a good deal more complicated. The section concludes with a result revealing the distinguished s-hierarchical nature of 'Lead (No)', a canonical subgroup of No's multiplicative group which plays a pivotal role in the aforementioned assignment of names to numbers.

3.1. Every surreal number has a canonical proper name. In order to define the Conway name of a surreal number we require a number of definitions beginning with the following classical ones from the theory of ordered groups.

Two elements a and b of an ordered group A are said to be Archimedean equivalent, written a b, if there are positive integers m and n such that m Ia > Ib and n Ib > Ia ; if Ia < Ib and it is not the case that b, we write IaI << Ib and a is said to be infinitesimal (in absolute value) relative to b; the class of all members of A that are Archimedean equivalent to some member of A is said to constitute an Archimedean class of A.

DEFINITION 11. An element of an s-hierarchical ordered group A will be said to be a leader of the group if it is the simplest member of the positive elements of an Archimedean class of A; by 'Lead (A)', we mean the class of all such leaders of A, and if x c Lead (A), then by 'LS(X)' and 'Rs,(x) we mean the sets Lead (A) n Ls(x) and Lead (A) n Rs(x), respectively. Finally, by 'L*x) and 'R*x) we mean the classes of all y C A+ such that y z for some z in Lsl(x) and RsI(x) respectively, where here and henceforth A+ denotes {a c A: a > O}.

Since the subclass of positive elements of an Archimedean class of an ordered group A is convex, the concept of a leader is well-defined. Some of the salient features of leaders are summarized by the following result.

THEOREM 10. If A is a nontrivial s-hierarchical ordered group, then: (i) {f0} is the simplest leader in A; (ii) for each x c A - {0}, there is a unique y C Lead (A) where y x; (iii) x c Lead (A) if and only if x ={O,L *(x) R*(x)}; (iv) (Lead (A), <A I Lead (A), <s I Lead (A)) is a lexicographically ordered binary

tree; (v) x c Lead (A) if and only if x = { Lsl,(x) I Rs,(x) } Lead(A); (vi) Lead (A') =


Lead (A) nA' ifA' is an initial subgroup of A; (vii) ifA' is an initial subgroup of A, then (Lead (A'), <A' I Lead (A'), <s, I Lead (A')) is an initial subtree of (Lead(A), <A Lead(A), <s I Lead(A)).

PROOF. Since (i) and (ii) are trivial, (iv) implies (v), and (vi) implies (vii), we consider (iii), (iv) and (vi). Let x c Lead (A). Then {0}UL,(U ?<< {x} << R? and if y={0 Lsl(x) Rs*(x)} and y #4 x, y c Lead (A) and y <s x. But then y c L*,(x) U R(x), which is impossible; and so y = x {0, L, (x) R , (x) }. Also, if x {,Lx) I R*(x)}, then {0} U L> (x) < {y c A+: y x} < R *(x); and so, since x c {y c A+: y x} C {z: {O} U L*x) < {z} < R*(x)}, x is the

simplest y c A+ such that y - x, proving (iii). Now let x, y C Lead (A) where x < y. If x is incomparable with y, then (by Definition 1) there is a z C A such that x < z < y where z <s x and z <s y. But then a <s x and a <s y where a is the unique member of Lead (A) such that a z; moreover x < a < y. On the other hand, if a c Lead (A) and a is a common predecessor of x and y such that x < a < y, then (by Definition 1) x is incomparable with y; thereby proving (iv). Finally, let x c Lead (A') and let y be the unique leader in A such that x y. Then y <s x; and so y c A' since A' is an initial subgroup of A. But then x y since x is the simplest member of the Archimedean class of A' containing y; and so x c Lead (A) n A'. Similarly, if x c Lead (A) n A' and y is the unique leader in A' such that x y, then y <s x. But then x = y since x is the simplest member of the Archimedean class of A containing y; and so x c Lead (A'). -]

For use in the proofs of the next group of results, a subset A of No is said to be cofinal with (coinitial with) a subset B of No if for each b c B there is an a c A such that a > b (a < b). Moreover, henceforth, 'nL' and j nR' are understood to denote typical members of the sets of all elements of the forms na and 1 b, respectively, where a c L, b c R, and n is a positive integer; and 'nwL',

ln CR, 'nC(t) I 2ln , etc., are understood to denote typical members of the analogously defined sets.

LEMMA 3. Let L and R be subsets of Lead (No) where L < R. Then {L I R}Lead(No)

exists and equals { 0, nL I R}

PROOF. Since {0}U{nL} and {1R} are subsets of No where {O}U{nL} < { 1 RI, there is (by virtue of Theorem 5) a y C No such that y {0, nL |R .} But since y c A {a C No+: a y} and {0} U {nL} < A < {f 1R}, y is the simplest

member of A and, hence, y c Lead (No). But {nL} is cofinal with L and {21 R} is coinitial with R, and so L < {y} < R. Moreover, since for any z c Lead (No) such that L < {z} < R, we have {0} U {nL} K {z} K {< R}, it follows that

y {L; R}Lead(No) q

In virtue of Theorem 10(iv) and Lemma 3, we now have THEOREM 11. (Lead (No), <No I Lead (No), <s I Lead (No)) is a complete lexico-

graphically ordered binary tree. Accordingly, the unique s-hierarchical mapping from Lead (No) to No (henceforth, IDNo) is a bijection.

As Theorem 12 below shows, the following recursive definition which is a variation on an idea due to Conway [7, p. 31] provides a unique appellation for each leader in No.

1246 PHILIP EHRLICH

DEFINITION 12. For each y c No, ogy = {, noLs(Y) | oy) }NO

THEOREM 12. coy = ID-1 (y) = t0,n(L | 21nw(R} for each y C No and each

ordered pair (L, R) such that y = {L I R}INo. And so, since (D-1 is s-hierarchical,

Lsl(cl') = o{WX: x C Ls(y)} and Rsl(,Y) = fox x C Rs(y)}.

PROOF. Plainly, coo = ID7o (0) = { 0}. Now suppose y c No - {0} and cox iD-1 (x) for all x c Ls(y) U Rs(y). Then, since by Theorems 10(v) and 11

(D~~ ~-1 1y ~D'():xCL~~ (- x LeadNo

ONo () {No (X :t~)}|{No ()XC s(y)}}

we have

-i CL

(y) - { CD |Rs y) }Lead(No)

And so, by Theorem 10 (iii) and the fact that {0} U {nwoLs(y) } is cofinal with L*y)

and { 1oRs(y) } is coinitial with Rs*l, we have

(D (y) {0, ncowLs(Y) |1 CoRs(y) } cy

But, by Theorem 2, if y = {L IR}N, then L is cofinal with Ls(Y) and R is coinitial

with Rs(Y), from which it follows that { 0, nL} is cofinal with {0, nwLs(y) } and

{?w1n~)R} is coinitial with { _)R5'(Y)}. But then coy = {0,nwOL (OR}N, since

{0, nw L} < {WCoY} < { R (OR

Since No is a vector space over R and every member of No - {0} is Archimedean equivalent to exactly one member of {coy: y C No} C No+, a familiar classical

argument (cf. [19, Lemma 2.4]) leads to the following

SIMPLE RESULT. If x c No - {0}, then there is a unique r C R - {O} and a unique

y c No for which Ix - rcoy << coy. This elementary observation plays pivotal roles in both the formulation and proof

of the following recasting of an argument sketched by Conway [7, pp. 32-33].

THEOREM 13. For each x c No - {0}, there is a unique descending chain Ix, a < fi c On, of convex subclasses of No-henceforth called a Conway chain whose

intersection contains x as its simplest member, and whose components are defined by

recursion as follows: I x is the subclass of allmembers of No oftheform s0 +rjo(Ya +a0 where

SO, = -0,if = 0

(i) sc, is the simplest member of nighJ ix, otherwise;

(ii) rjcoYa henceforth called the a-term of x is the unique member of No for

which rc, c R - {0}, y,, c No and Ix - (s,, + rjcoY)I << wYac;

(iii) I ac I << CoYa .

Furthermore, if (yc)c<fe n is a nonempty descending sequence of members of No

and (rcX)c,<fl is a sequence of member of R - {0}, there is an x C No - {0} whose

Conway chain Iox, a < fi, is characterized by the condition that rjcoYa is the a-term

of x for each a < fi.


PROOF. If x c No - {0}, there is a unique ro c R - {0} and a unique yo C No for which Ix - (s0 + rocoy)I <?< coy'; and so 1o, which is obviously a convex subclass of No containing x, is uniquely determined. Now suppose that for each a < some y > 0 we have already defined the ath-component, I,, of a descending chain of convex subclasses of No whose intersection contains x, and further suppose sy is the simplest member of nF < y I. If x - sy = 0, then 3 = y, the descending chain is determined, and x = s# is the simplest member of its intersection. On the other hand, if x - sy :4 0, there is a unique ry c R - {0} and a unique yy c No for which Ix - (sy + rycYY ) I <W< wY?; and so Iy, which is a convex subclass of No containing x, is itself uniquely determined. Accordingly, to complete the first part of the proof we need only note that the above process can only be continued over all (x < some /3 c On since prNo (x) is a set and, in virtue of the above construction, (s,),,< is a chain in prNo (x). Turning to the second part, let (y11f)a<fE~n and (ra)O,, have the properties specified above, and let I,, a < be the descending chain of convex subclasses of No where I, consists of all members of No of the form sa + racoYa + ac,

where sa and ac, are defined as in (i) and (iii) above. Since No is a complete s- hierarchical ordered vector space over R, the descending chain Ic, a < /3, of convex subclasses of No exists, as does (by virtue of Theorem 4) the simplest member of the intersection of the chain, say, x. To complete the proof it only remains to draw attention to the simple fact that Ix - (so, + racoY)I << CoYa for each a < /3. 1

As we have already noted, Definition 12 assigns a unique appellation to each leader of No. The completion of the assignment is now carried out via

DEFINITION 13. We will refer to the formal expression

Zcow. rc,

as the Conway name of a surreal number x, treat the Conway name of x as a proper name of x and, accordingly, write 'x = ea<: CoYa . rc,' if either x = 0 and /3 = 0, or x is the simplest surreal number whose oa-term is racoYa for all a < fi. (Moreover, for 3 = 1 we will delete the summation sign when convenient).

In virtue of Definition 13 and Theorem 13 we now have the following version of Conway's Normal Form Theorem [7, p. 33].

THEOREM 14. Every surreal number has a Conway name; distinct surreal numbers have distinct Conway names; furthermore, the formal expression

Z woya. rc, a<,B

is the Conway name of some surreal number namely, the simplest surreal number in { ,<f JoYa . rc, + a: jal << CoYa for all a < fi} if and only if (Ya),<E0n is a

(possibly empty) descending sequence of members of No and (rc)c,<f is a sequence of members of R - {0}.

As the reader will have noticed, despite its use of summation and product signs, the Conway name of a surreal number x is not really a sum of products at all but rather a formal expression that denotes x as characterized by the corresponding definite description specified in Definition 13. On the other hand, as the following

1248 PHILIP EHRLICH

simple consequence of Theorem 13 together with Definition 13 makes clear, the use of the summation and product signs are as appropriate as they are revealing.

BASIC OBSERVATION. coy . r = rowy (i.e., = r No coY) for each y C No and each r c R - {fO}, andfor each nonempty descending sequence (yO4f<pe 0, of members of No and each sequence (rO4f</ of members of R - {fO}

S wa. ro=e WYa. ro+No 5 C row a<:l aO<p vc<a<:

where '+No' and '.No' designate the standard field operations in No. 3.2. The s-hierarchical ordered field No revisited. There are a number of signif-

icant relations between surreal numbers that are reflected in terms of relations between their respective Conway names. The following three such examples (which are essentially due to Conway and Gonshor) play important roles in the subsequent discussion.

THEOREM 15. (i) 1< ctYa roe <S wo<fl ctYa . ro, whenever , </:; (ii)

E co . ro{Z ). rc( + cy.( r 2n)

No

Icow . r + co (rf + 2n ) a<,B O<n<co

(iii)

IcoY r.g={A t rog + (op rU 2n)

No

5 c ro, + coy. (rU + 2n) }O < a<,u O<n~co,/u<#

if f is a limit ordinal (where the inequalities 0 < n < co and ,u < indicate that n and u range over allpositive integers and all ordinals < /3, respectively).

PROOF. (i) follows from Definition 13 together with Theorem 13. And, in virtue of the above Basic Observation and rudimentary algebra, an element x of No lies between the sets of left and right members in the expression in (ii) if and only if x = C,<+I CoYa . rc, + z for some z where IzI << woYa for all a < p + 1. Thus, by the second part of Theorem 14, (ii) follows. The proof of (iii) is similar. -]

DEFINITION 14. A surreal number x will be said to be an approximation (a strict approximation) of a surreal number ,<fcoYa . rc, if x = ,<, oYa . rc, for some a < /3 (a < /). If every approximation of a member of a class of surreal numbers is itself a member of the class, the class will be said to be approximation complete.

As was noted above, the following result, in which the operations +H and *H

are defined on, and the relation <H is defined between, members of No denoted


by their respective Conway names is inspired by a result due to Conway. Here and henceforth, for the sake of descriptive simplicity, we permit the concept of the Conway name of a surreal number to be naturally expanded so as to allow for the insertion and deletion of "dummy" terms with zeros for coefficients.

THEOREM 16. (No,+H, H <H, <s) = (No, +,*, <, <s) when +H, H and<H are defined by the following conditions where terms with zeros for coefficients are understood to be inserted and deleted as needed:

Z, woY . ay +H i, .coy.

by coy. (ay + by), yENo yENo yENo

Z WY. aY H Z i WY. by =Z Y. a v yENo yENo yENo (ji,v)ENo x No

>EyENo cY . ay <H >yENocoy . by, if ay = by for all y > some x c No and ax < by.

PROOF. In virtue of Theorem 14, to establish the theorem we need only show that u <H v whenever u < v, and that +H and H satisfy the s-hierarchical constraints on sums and products. To prove the latter, we will rely upon the fact that (No, <H, +H, *H) is an ordered field when <H, +H and H are defined as above, a result that follows immediately from Theorem 14 and a classical field-theoretic argument of Hahn [18].

Let u = EYENocoy . ay and v Y EYeNoo . by where terms with zeros for coefficients have been inserted where needed, and further suppose u < v. Since u :4 v, it follows from Theorem 14 that there is an x c No such that ay = by for all y > x and ax :4 bx. Moreover, by virtue of Theorem 13 and the assumption that terms with zeros for coefficients have been inserted if needed, there is an I c Lead(No) andthereare j, k c No where J j, Jkl << I such that axwOx = axl +j and bxcox = bxI + k. But since u < v, it follows that axI + j < bxI + k and, hence, that ax < bx since I jI, IkI << 1; thereby showing u <H V.

Now suppose x,y C No. Then x = {Lx I Rx} and y {LY RY} where Lx, Ly, Rx and RY are of the forms specified in Theorem 15. Also let Lx+HY =

{xL +H Y, X +H yL} where XL c LX and yL c Ly, and suppose Rx+HY, LX.HY

and Rx.HY are defined analogously, the latter two in concord with the constraint on products from Definition 3. By virtue of Theorems 3.2 and 3.5 of [17], to show that +H and H satisfy the s-hierarchical constraints on sums and products, it suffices to prove this using the just-cited representations of x and y. Accordingly, beginning with addition, we may argue as follows.

Since (No, +H, <H) is an ordered group, it follows from the argument given following the statements of Definitions 2-4 that Lx+HY <H {X +H Y} <H RX+?HY

Moreover, if both x = 0 and y = 0, the result obviously holds. Suppose then that either x 4 0 or y 4 0. Then x +H y has a strict approximation. Moreover, if a is a strict approximation of x +H y, then by appealing to the definition of <H and the contents of the Lxs, Ly s, Rx s and RY s it is a routine matter to show that either there is a b c Lx+HY such that a < b or there is a b C Rx+HY such that b < a. Consequently, xL +H Y <H Z <H XR +H y if and only if z =(x +H Y) +H W where

1250 PHILIP EHRLICH

x+HYis an approximation of (x +H y)+Hw. But then x +Hy <s (x +H y)+Hw,

and so we have x +H y ={Lx+HY Rx+HYN

The argument for multiplication is similar except here the trivial case is where either x = 0 or y = 0, and one notes that since (No, +H, -H <H) is an ordered field, it follows from the argument given following the statements of Definitions 2-4 that

LX.HY <H {X H Y} <H Rx.HY.

By appealing to part (iv) of Theorem 10 together with Theorems 12 and 16, it easy to see that the s-hierarchical mapping ID-1 from No onto Lead (No) is an isomorphism of ordered groups, and hence that

THEOREM 17. (Lead (No), <No Lead (No), <s I Lead (No), Lead (No), 1) isa complete s-hierarchical ordered group (written multiplicatively).

?4. The spectrum of s-hierarchical ordered fields. Theorem 18 below provides necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No. To formulate both the theorem and a proposition employed in our proof thereof, we require the following definitions.

DEFINITION 15. Let R (G) On be the ordered field of power series (with order, addition, and multiplication defined a' la Hahn (see Theorem 16)), consisting of all formal power series of the form

E ratYa

a<fl

where (ya)a<fCOn is a (possibly empty) descending sequence of elements of an

ordered abelian group G and ra Ec R - {O} for each a < fl where R is the ordered field of reals. If the universe of G is a set, R (G)On is the familiar Hahn field (with exponents in G), henceforth denoted R (G).

DEFINITION 16. An element x of R (G)On will be said to be a truncation (a proper truncation) of E<, ratYa c R (G)On if x = a<, rtYa for some u < ,6 (u < ,6 ); if every truncation of a member of a subclass of R (G)On is itself a member of the subclass, the subclass will be said to be truncation complete. A subfield F of R (G)On will be said to be cross sectional if {t9: g E G} c F (i.e., if {t9: g E G} is a subgroup of the multiplicative group F* = {x E F: x > 0}). Moreover, if F is a truncation complete, cross sectional subfield of R (G)On, then the ordered subfield RF = {r EE R: rt? E F } of R will be said to be the coefficient field of F.

Using Definitions 15 and 16 it is a straightforward matter to prove PROPOSITION 6. If F is a truncation complete, cross sectional subfield of R (G)On,

then all of the coefficients in the members of F come from RF, and { E,,<v rp tYP G

F: v is an infinite limit ordinal and rO 1 } U {t9: g E G} constitutes a class of generators for F considered as a vector space over RF.

By Theorems 14 and 16, No is isomorphic to R (N?)On a result first established in [11]; also see [3]. We now prove more generally

THEOREM 18. An ordered field is isomorphic to an initial subfield of No if and only if it is isomorphic to a truncation complete, cross sectional subfield of a power series field R (G) Onwhere G is isomorphic to an initial subgroup of No.


PROOF. Let A be an initial subfield of No. To prove the "only if" portion of the theorem it suffices to show A is an approximation complete subfield of No (see Definition 14) where G = {y: coy E Lead (A) } is an initial subgroup of No for then plainly { E< rtY- c R (G)0,1 : w<,Y . r, Ec A} is a truncation complete, cross sectional subfield of R (G)0,, that is isomorphic to A where G has the desired form. But by Theorems 14 and 16, No is approximation complete, and so by Theorem 15(i), A is likewise approximation complete. Moreover, since A is an initial subfield of No, coo = 1 E A and coX -oy, (coX) l E Lead (No) n A for coX, coY c Lead (A), and so (by parts (i), (vi) and (vii) of Theorem 10) co, coX

cy, (coX)-l E Lead (A) and Lead (A) is an initial subtree of Lead (No). But this together with Theorem 17 implies Lead (A) is an initial subgroup of Lead (No). But since G is the image in No of the unique s-hierarchical embedding from Lead (A) to No, G is an initial subgroup of No. Turning now to the converse, let F be a truncation complete, cross sectional subfield of the power series field R (G ) 0, where G is an initial subgroup of No. Then F is isomorphic to an ordered subfield A of No consisting of all elements having Conway names of the form <, coya . ra where

Za<p ra tYa c F. Accordingly, to show that A is an initial subfield of No, it suffices to show that KA, <, I A) is an initial subtree of KNo, <s). It is to this end that we now turn our attention.

Let ao, . . ., aa, . . . (a < /3) be a well-ordering of G such that PNo (a/X) < PNO (a,,) whenever 1u < v < /3. Clearly, such a well-ordering can be constructed and for any such ordering a0 = 0 {0 1 0} and, more generally, by Theorem 2, for each a < /3, a,= {La I Ra},NO where (La, Ra) is the unique partition of {a6: 3 < a} such that La < {aa,} < Ra. Furthermore, it is easy to see that A considered as an ordered vector space over the coefficient field RF of F is equal to Ua<p Aa

where Aa is the subspace of A consisting of 0 together with all of the elements of A having exponents in G {a= : 3 < a}. So, since KAo, <s I Ao) = (RF <s |RF)

K{o09 r: r E RF}, <s {0 * r: r E RF}) is (by Theorem 8) an initial subtree of

No, and since the union of a chain of initial subtrees of No is itself an initial subtree of No, to establish the theorem we need only prove: if 0 < a < /3, then KAa, <s A Aog)

is an initial subtree of No whenever KU,,<a A,, <S I U,<a A, is an initial subtree

of No. Let 0 < a < /3 and suppose KU,,<a A,,c <s I U,,<a A,) is an initial subtree of

No. Also let

Za {= E CYu . ri c Aa - U A,: v is an infinite limit ordinal and ro = 1} ,c<V v ic<a

and suppose bo, . . ., b, . . . (u < -) is a well-ordering of Za U {(car } such that for all y < 3 < -, by precedes b& in the well-ordering if and only if the initial sequence of ordinals over which the exponents in by are indexed is contained in the initial sequence of ordinals over which the exponents in by are indexed. By appealing to Proposition 6 (and recalling that (X)A denotes the subspace of A generated by X) it is easy to see that Za U {coac} Uic<a A,1 is a set of generators for Aa, from

whence it follows that Aa = U,<, B, where Bo ({bo} U Usi<a A)j A and B, =

1252 PHILIP EHRLICH

({bu} U U6<, B6)A if 0 < u < T. Using this fact and the fact that b, V U6<u B6 whenever u < I, we now show that Aa is an initial subtree of No by showing that B, is an initial subtree of No for each u < -c.

To begin with, notice that bo = co={LlRa}No where (La, R,) is the unique partition of {a6: 8 < a } such that La < {a,} < Rot It therefore follows from Theorem 12 that

bo = co- ={0, ncoLa Ic Ra}

where n ranges over the positive integers. But since every element of U, <a A,, - {O} is Archimedean equivalent to a unique member of {w a'5: 8 < a} C U,,a, A,, - {0}, it follows that there is a unique partition (Lb, R6) of U,<<a A,, where Lo < R' such

that {0, ncoLa } is a cofinal subset of Lo and { I CORa } is a coinitial subset of R.

But then, by Theorem 2, bo {Lo I R6}No; and so, by Theorem 6, Bo is an initial subtree of No.

If Za = 0, then Bo = Aa and, as such, we are finished; if not, let 0 < U < T

and, as our induction hypothesis, suppose U6<, B6 is an initial subtree of No. Since 0 < u < i, b, has a Conway name of the form Ea< coa . r,, where m is an infinite

limit ordinal and ro = 1. Moreover, by Theorem 15, bu = {L I R}INo where

L Ea~p ra + co' (r,, - {ER<LC ( 2n ) W~~~~O<n<co,pu<7

and

R c E o ra + co. (r. + 2) }

But since, by construction, L U R C U6<, B6 and bo V U6<< B6, there is a partition

(L', R') of U6<, B6 such that

bu = {L' I R' I}?.

Therefore, by virtue of the induction hypothesis and Theorem 6, Bow is an initial subtree of No. Thus, by induction, B, is an initial subtree of No for each u < I, and so Aa and, hence, KA, <, I A) are initial subtrees of No; thereby proving the theorem. -1

Every Hahn field is of course a truncation complete, cross sectional subfield of a Hahn field; and the existence of other important classes of truncation complete, cross sectional subfields of Hahn fields has been known for some time now (cf. [1], [18]). Moreover, by a straightforward application of the constructions introduced by Rayner [25] and Ucsnay [28], one can obtain for each s-hierarchable ordered group G a vast array of Henselian, truncation complete, cross sectional subfields of R (G) On In addition, making minor adjustments in a recent argument of Mourgues and Ressayre (and Delon) ([22], [23]), we can prove

PROPOSITION 7. Every real-closed ordered field is isomorphic to a truncation

complete, cross sectional subfield of a power series field R(G)On where G is a di-

visible ordered abelian group.


PROOF. Let A be an ordered field whose universe may be a proper class, K G, ,<,0) be an ordered abelian group, and

A (QA) = { cE A: -- r < x < rfor some r E QA}

where QA is the rational subfield of A. By a cross section of A we mean a subgroup {xg: g E G} of the multiplicative group A+ in which every positive element of A is Archimedean equivalent to exactly one such xg and xg x = xo9 for all g, g' E G; and by the canonical residue field of A we mean the ordered subfield of R that is the image of the ordered ring homomorphism p: A (QA) -> R defined by the condition: p (a) = sup {r C Q: p'`(r) < a} for each a c A (QA).6 When A has a cross section {xg: g E G}, it is an invariant of A that is isomorphic to G; and, as is well known, if A is a real-closed ordered field, then A has a divisible cross section and contains an isomorphic copy of its canonical residue field, the latter of which is real-closed. Using these familiar results, Mourgues and Ressayre have established the said proposition for real-closed ordered fields whose universes are sets ([23, p. 647: Corollary 4.2], [22, p. 257: Corollary 4.2]). Relying on their theorem and their proof thereof, the result may be readily extended to structures whose universes are proper classes using the following elementary

PROPOSITION. Let A be a real-closed orderedfield whose universe is a proper class, RA be an ordered subfield of A that is isomorphic to the canonical residue field of A, and {xg: g E G} be a cross section of A. Also, let xO = 1 E B where B is a basis for {xg: g E G } considered as an ordered vector space over the rationals, and let (xga)a<0n be a well ordering of B - {x0}. Then A is the union of a continuous chain Aa, a < On, of ordered real-closed subfields of A (whose universes are sets) where AO = RA, andfor each a < On, A,+, is the maximal subfield of A that is an Archimedean extension7 of the real-closed subfield of A generated by A, U {Xf }.

Thus, by combining Proposition 7 with Theorems 9 and 18 we have

THEOREM 19. Every real-closed ordered field is isomorphic to an initial subfield of No.

?5. Ordinals and omnific integers. Conway [7] isolated two substructures of No- one a semiring and the other a ring that he dubbed No's ordinals and omnific integers, respectively In this section we generalize these conceptions and apply them to s-hierarchical ordered groups and s-hierarchical ordered fields, more generally. We begin by limiting our attention to initial substructures of No.

DEFINITION 17. Let A be a nontrivial initial subgroup of No. An element x of A will be said to be an omnific integer of A if x = {x - 1 I x + 1}A, it being understood that 1 = {o | A . Henceforth, by Oz (A) we mean the subclass of all omnific integers of A with the induced order.

6The above definitions of cross section and canonical residue field replace the more traditional definitions of cross section and residue field employed by Mourgues and Ressayre, the latter of which are not well-defined notions in NBG for fields whose universes are proper classes.

71f G and G' are ordered groups where G C G', then G' is said to be an Archimedean extension of G if for each a e G'+, there is a b e G+ and positive integers m and n such that na > b and mb > a.

1254 PHILIP EHRLICH

The nature of the system of omnific integers of a nontrivial initial subgroup of No is greatly clarified by the following theorem, the nonroutine portion of which follows immediately from Conway's proof of Theorem 31 of [7].

THEOREM 20. If A is a nontrivial initial subgroup of No, then Oz (A) is the subclass of all x c A such that

x Zcow . ac, a<p

where (ya)a,<p is a (possibly empty) descending sequence of nonnegative members of No and aa, is a nonzero integer if yc = 0; and so, if A is a nontrivial initial subgroup (initial subfield) of No, then Oz (A) is a discrete subgroup (subring) of A in which for each x E A there is a z c Oz (A) such that z < x < z + 1.

DEFINITION 18. An element a of an initial subtree A of No will be said to be an ordinal if a has a representation of the form a = {L I 0} (i.e., if and only if a is a (possibly empty) sequence of +s). By On (A) we mean the subclass of all ordinals of A.

THEOREM 21. On (A) which is an initial subtree, as well as a cofinal subclass, of A is a well-ordered class in which for each a c On (A), a = {ordinals y < a I }; if A = No, then for every subset S of On (A) there a member of On (A) greater than every member of S; if A is a nontrivial s-hierarchical ordered group (s-hierarchical orderedfield), then On (A) is a subsemigroup (subsemiring) of Oz(A).

PROOF. The assertion between the dashes is obvious, and the remainder of the first two parts of the theorem follows from an argument of Conway [7, p. 28]. Moreover, if a, f, c On (A) where A is a nontrivial initial subgroup of No, then

a - 1 > each ordinal y < a and a + 1 < each y c 0, which, together with the

fact that a = {ordinals y < aI 1}. implies that a = f a- 1 Ia + 1}A and, hence,

that On(A) C Oz(A). Finally, since a = jOaL I} and /l = ,#L 1a , a + 6 =

{aL + /i, a + /iL 11}A On (A) and, if A is an s-hierarchical ordered field, in which

case a/i is well defined, a/= {a L/ + a/iL - aL/L I}A c On (A). -

In virtue of Theorem 21 there is a one-to-one order preserving mapping between No's ordinals and the ordinals defined in any of the more usual fashions. This being the case. we may adopt the convention that identifies the class On of all ordinals (that are sets) with the class On (No) rather than with the class of von Neumann ordinals as we have heretofore supposed. With this convention henceforth assumed to be in place. the familiar Cantorian operations on ordinals written a +c /i, a . f,, and ao #-as well as the following classical definitions and results based thereon are understood to be formulated in terms of the members of On = On (No). Moreover, for the sake of convenience, henceforth we will extend the use of the term 'ordinal' to include On = (xa)aCOf,, where xa= + for all a c On, extend the ordering on

On to On U {On} in the expected manner and, on occasion, write On = cow, the expression on the right being the "Conway name" of the "leader" {0, nLsonL s } in No U {On}.

As the reader will recall, an ordinal number a is said to be (additively) indecomposable if /i +, y < a whenever /i. y < a. Moreover, if 0 < a < On, then a is indecomposable if and only if a = co' - for some ordinal number W < On. This


together with other familiar theorems concerning ordinals leads to the following well-known theorem due to Cantor: Every ordinal a < On has a unique Cantor Normal Form, i.e., a unique representation of the form

E09c *oa c aa

a<n

where (mOa<) is a (possibly empty) finite descending sequence of ordinals < On, (aa)a<n is a sequence of finite ordinals > 0, and the 'c' affixed to the summation sign indicates the sum is Cantorian.

By extending Cantorian exponentiation of ordinals to the indecomposable ordinal On, it is easy to see that On = cofc On and, hence, that co', On = C)On. The following theorem, which is central to the remainder of the paper, demonstrates that the theory of Conway names generalizes the theory of Cantor normal forms for classical ordinals as well. The proof makes tacit use of the well-known result that to compare ordinals written in Cantor normal form one compares them by first differences [e.g., 20, p. 127].

THEOREM 22. Ec0<n wct -c a, = >?,<n c a . a, whenever (Wpa<) is a (possibly empty) finite descending sequence of ordinals < On and (aa)<n is a sequence offinite ordinals > 0.

PROOF. Since the class of ordinals covered by the theorem is On itself, we pro- ceed by induction on T E On. For T = 0, both sums are the empty sum. Now let T > 0 and suppose the result holds for all y < T. Since T > 0, -c

Za<n=m+1 c c a, = Zc<m w0 cC ac, +c cl)-w

, *c am for some m < c; and

so, by the definition of +c, C sup? {EZ<m w'O c a, +c /: ,6< CO w'cn * am}. But then -c {EZ<m w) c C aa + /3: ,c < C) v'c am }; and this together with

the induction hypothesis implies -c = {Ea<m w0 a, + ,6: /3 < 0wn- . am I }. Now

let I' = Ea<m w)- . a, + cwe- . am. Then, as the proof of Theorem 21 demon-

strated, -c' = {E<mW00% . aa + fl,X+ 0a - . am I} where /3 < Ho . am, A <

Za<m w0- . aa, and fl, A E On. But since c~t0n-Iis indecomposable and 07-I >

o_ * am, it is easy to see that A + wfn7

. a)" < Za<m

apt . a, for all ordinals

i < Za<m 0D0 a,. But then - {Ea<m WC) . aa + fl: fl < oPR' . am }

Za<m+1=n ouk. a, A

In virtue of Theorem 22 and the preceding remark concerning the comparison of ordinals written in Cantor normal form, we now have

THEOREM 23. There is a unique order preserving mapping f from On onto the ordered subclass of all x E No such that x = >j3<n w0)'P . ac, for some (possibly empty) finite descending sequence ((pa,) < of ordinals and some sequence (aa)a<n of

finite ordinals > 0; furthermore, f (a) = a for each a Ec On.

The next definition is motivated by Theorem 23 and the classical Archimedean condition.

DEFINITION 19. An s-hierarchical ordered group A will be said to be a-Archimedean if a is the height of On (A) (considered as a subtree of A); moreover, if A is an s- hierarchical ordered field, a E A, q is an ordinal, and h: On (A) -> On is an s-hierarchical mapping, then we define qa = h (q) a if q E h (On (A)), qa being undefined otherwise.

1256 PHILIP EHRLICH

It is evident that every s-hierarchical ordered group is a-Archimedean for some a. Also evident is

THEOREM 24. Let A be a nontrivial s-hierarchical ordered group and h: On (A) -> On be an s-hierarchical mapping. A is a-Archimedean if and only iffor each a E A there is an q Ec On (A) such that -a < a < q where h (q) < a. Moreover, if A is an s-hierarchical ordered field, then A is a-Archimedean if and only iffor all a,b E A+ where a > b there is an q < a such that qb > a; furthermore, A is Archimedean if and only if A is co-Archimedean.

By appealing to Theorem 23 together with the definitions of +H and H as stated in Theorem 16, it is easy to see that an initial sequence {a,: a < Pl < On} of ordinals is closed under addition (multiplication) if and only if P = w0 for some ordinal (indecomposable ordinal) p < On. Accordingly, if for each ordinal (indecomposable ordinal) p < On we let SG (aLP) (SR (ai)) be the ordered class of all ordinals less than w0 with sums and products defined as in Theorem 16, then the SG (aiP)s (SR (0i)s) so defined are the only subsemigroups (subsemirings) of No that consist of initial sequences of ordinals of No.8 And this together with Theorems 21, 23 and 24 and the fact that for each s-hierarchical ordered structure A there is one and only one s-hierarchical mapping hA: A -* No, hA being an s-hierarchical embedding, we have

THEOREM 25. Every nontrivial s-hierarchical ordered group (s-hierarchical ordered field) A is w0-Archimedean for some nonzero ordinal (nonzero indecomposable ordinal) p < On; and if A is an w0-Archimedean s-hierarchical ordered group (s- hierarchical ordered field) then A contains a cofinal, canonical subsemigroup (subsemiring) On (A) called the ordinal part of A that is isomorphic to SG (00)

(SR (0P)) and which consists of all a Ec A such that a = {L I 0}A for some L C A; On (A) in turn is contained in a discrete, canonical subgroup (subring) Oz (A) of A called the omnific integer part of A consisting of all x E A such that x = {x - e I x + e}A where e is the simplest positive element of A; in addition, for each x E A there is a z E Oz (A) such that z < x < z + e.9

?6. Concluding remarks. While laying the groundwork, the present paper merely takes a first step in the development of a general theory of s-hierarchical ordered algebraic structures. We conclude by mentioning a few directions for further research that the author (and, hopefully, others) will turn to in the not too distant future. To begin with, in addition to shedding further light on the nature of the s-hierarchical

8As the reader will notice, for each ordinal A, SG (co9') is essentially the ordered semigroup of all ordinals (written in Cantor normal form) less than cow with order defined by first differences and addition defined a la Hessenberg [cf. 27], and for each indecomposable ordinal A, SR (cow') is essentially the ordered semiring that results from supplementing SG (co9') with a Hessenberg product [cf. 27]. Unlike the SG (co9')s, thus construed, which have been discussed in the literature [26], the SR (co9')s appear to be new (except for the cases where the cow s are regular initial numbers [27]).

9The omnific integer part of an s-hierarchical ordered field is an integer part in the sense of Mourgues and Ressayre [23], i.e., a discrete subring I of an ordered field A in which for each x E A there is a z E I such that z < x < z + 1. However, while every s-hierarchical ordered field has an integer part, there are ordered fields having integer parts that do not admit relational extensions to an s-hierarchical ordered field each Hahn field JR (G) where G does not admit a relational extension to an s-hierarchical ordered group is an example of such a system.


ordered algebraic structures we have investigated thus far, it remains to investigate their natural generalizations the s-hierarchical ordered algebraic structures that are defined by replacing the references to ordered groups, ordered fields and ordered K-vector spaces in Definitions 2-4 with references to ordered semigroups, ordered rings, and ordered K-modules, respectively. While some of our results such as Lemma 2 and Theorem 5 as well as their proofs carry over to these more general classes of s-hierarchical ordered structures, their isolation and analysis remain wide open. Similar remarks apply to s-hierarchical analogs of ordered semirings and ordered K-semimodules, which also deserve attention. In addition, unlike the ordered fields of reals and surreals which admit (up to isomorphism) only one relational extension to an s-hierarchical structure, many s-hierarchable structures admit relational extensions to nonisomorphic s-hierarchical structures and an analysis of the classes of such alternative extensions also remains to be provided. Finally, while we have devoted the bulk of our attention to s-hierarchical ordered structures per se, detailed analyses of the hierarchical relations that exist between their respective elements also remain to be provided including a detailed analysis of the tree-theoretic relations that exist between surreal numbers denoted by their respective Conway names.

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