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

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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, +,
NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES 1241 ?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
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Number Systems with Simplicity Hierarchies: A ... - Ohio University

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