Number Systems with Simplicity Hierarchies: A ... - Ohio University

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 spacesas
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].