Number Systems with Simplicity Hierarchies II

TL;DR

This paper extends the analysis of the simplicity hierarchy of surreal numbers to ordered abelian groups and domains, characterizing their initial substructures and their relation to Conway's surreal number field No.

Contribution

It establishes new characterizations of initial subgroups and subdomains of No, including their convexity and discreteness, and identifies the theories where all models are initial substructures of No.

Findings

Initial subdomains of No are discrete iff they are initial subdomains of Oz.

Convex subgroups and subdomains of initial subfields of No are characterized.

Theories of divisible ordered abelian groups and real-closed ordered fields have all models as initial subgroups/subfields of No.

Abstract

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.