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

1246 PHILIP EHRLICH
DEFINITION 12. For each y c No, ogy = {, noLs(Y) | oy) }NO
THEOREM 12. coy = ID-1 (y) = t0,n(L | 21nw(R} for each y C No and each
ordered pair (L, R) such that y = {L I R}INo.
Lsl(cl') = o{WX: x C Ls(y)} and Rsl(,Y) = fox x C Rs(y)}.
And so, since (D-1 is s-hierarchical,
PROOF. Plainly, coo = ID7o (0) = { 0}. Now suppose y c No - {0} and cox
iD-1 (x) for all x c Ls(y) U Rs(y). Then, since by Theorems 10(v) and 11
(D~~ 1y ~-1
ONo ()
~D'():xCL~~
{No (X :t~)}|{No
(- x
()XC s(y)}}
we have
LeadNo
-i
CL
(y) - { CD |Rs y) }Lead(No)
And so, by Theorem 10 (iii) and the fact that {0} U {nwoLs(y) } is cofinal with L*y)
and { 1oRs(y) } is coinitial with Rs*l,
we have
(D (y) {0, ncowLs(Y) |1 CoRs(y) } cy
But, by Theorem 2, if y = {L IR}N, then L is cofinal with Ls(Y) and R is coinitial
with Rs(Y), from which it follows that { 0, nL} is cofinal with {0, nwLs(y) } and
{?w1n~)R} is coinitial with
{
{0, nw L} < {WCoY} < { (OR R
_)R5'(Y)}. But then coy = {0,nwOL (OR}N, since
Since No is a vector space over R and every member of No - {0} is Archimedean
equivalent to exactly one member of {coy: y C No} C No+, a familiar classical
argument (cf. [19, Lemma 2.4]) leads to the following
SIMPLE RESULT. If x c No - {0}, then there is a unique r C R - {O} and a unique
y c No for which Ix - rcoy