Number Systems with Simplicity Hierarchies: A Generalization of
Conway's Theory Of Surreal Numbers

Philip Ehrlich

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%TCIDATA{Created=Thu Jun 11 13:04:19 1998}
%TCIDATA{LastRevised=Fri Jun 12 10:17:51 1998}
\textbf{Number Systems with Simplicity Hierarchies: A Generalization of
Conway's Theory Of Surreal Numbers}
Philip Ehrlich
In his monograph \textit{On Numbers and Games,} J. H. Conway introduced an
ordered field $No$ consisting, as he aptly quips, of ``all numbers great and
small.'' In addition to its distinguished structure as a maximally inclusive
ordered field, however, $No$ has a rich algebraico-tree-theoretic structure,
or \textit{simplicity hierarchy,} that emerges from the recursive clauses in
terms of which it is defined. In the present paper, we introduce a novel
class of structures whose properties generalize those of $No$ and explore
some of the relations that exist between $No$ and this more general class of
\textit{s-hierarchical ordered structures} as we call them. More
specifically, we define a number of types of s-hierarchical ordered
structures (groups; fields; etc.) as well as a corresponding type of\textit{%
\ s-hierarchical mapping,} identify $No$ as a \textit{complete}
s-hierarchical ordered group (ordered field; etc.), 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 monomorphisms of their respective
kinds (ordered groups; ordered fields; etc.) whose images are initial
subtrees of $No$. This together with the completeness of $No$ enables us to
characterize $No$, up to isomorphism, as the unique complete as well as the
unique \textit{universal} and the unique\textit{\ non-extensible},
s-hierarchical ordered group (field, etc.). Following this, we begin the
process of uncovering the spectrum of s-hierarchical ordered structures.
Given the nature of $No$ alluded to above, this reduces to revealing the
spectrum of \textit{s-hierarchical substructures} of $No$, i.e., the
subgroups, subfields, etc. of $No$ that are initial subtrees of $No$. Among
our findings are: \textit{every divisible ordered abelian group is
isomorphic to an s-hierarchical subgroup of} $No$, and \textit{every
real-closed ordered field is isomorphic to an s-hierarchical subfield of} $No
$. We also generalize Conway's theories of \textit{ordinals} and \textit{%
omnific integers} by showing that every s-hierarchical ordered field $A$
contains a cofinal, canonical subsemiring $On(A)$ - the\textit{\ ordinal
part }of $A$ - which in turn is contained in a discrete, canonical subring $%
Oz(A)$ of $No$ - the\textit{\ omnific integer part }of $A$ - in which for
each $x\in A-Oz(A)$ there is a $y\in Oz(A)$ such that $y the multiplicative identity of $A$ - is the least positive element of $Oz(A)$%
. When $A$ is a substructure of $No$, $Oz(A)$ is a subring of $No$'s omnific
integers and $On(A)$ is a subsemiring of $No$'s subsemiring of all ordinals
(with sums and products defined naturally).