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

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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
Z NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES 1247 PROOF. If x c No - {0}, there is a unique ro c R - {0} and a unique yo C No for which Ix - (s0 + rocoy)I
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Number Systems with Simplicity Hierarchies: A ... - Ohio University

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