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

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