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

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 (shierarchical
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.