Further Results on Order Types and Decompositions of Sets
Seymour Ginsburg · Transactions of the American Mathematical Society · 1954
thermore, iff is any element of K~iA,) for which the power of the set EC\fiA,) is 2^°, then there are 21*0 elements x in Ai such that fix) is also in Ai.Proof.Since E has property C, it follows that for each element/in P*(P), the power of the set, [x|/(x)?ix, xGP}, is 2N°.Well order the elements of E and P*(P) into the two sequences, {x¡}, £<0, and {/{}, £<0, where each element in P*(P) appears 2^° times in the latter sequence.If a = co, let ß = a.If a is an integer let ß = a -1.Suppose that for each ordinal number £, where £ ¿,m|m</3} and by {^'m| w</3} the first ß2 elements in the set E-Y".Let A, = F" yj {pl'm\ i^2,m <ß] and P" = __" -[A^VJ f*iA")].Let ^'° be the first element in P" and p"° =fß[p^°).Since the power of X" is 2K° and the power of A» is <2t<l), the two distinct elements p3f and p*f certainly exist.In general, for m<ß let p^™ be the first element in the set P,-[G:V7/*(G;)], where G/» = {_V I * < OT} ^ (Í»• \i <m).Let/»*" =/,(#•").Let G = _% VJ {¿*,m | » ^ 4, m < /.} and P" = __" -[C" VJ/"*(G)].Let £°'° be the first element in P" and p5f=fliip6f).Let p^ and /#', 1 ui<ß, represent no elements.Denote by Z" the set Z" = {x|xGP, /M(x)GP}-Suppose that the power of Z" is 2N°.Let P" = Y" VJ {^"m | i Ú 6, w < 0} and F" = Z" -[P" VJ /*(P")].Let ^'° be the first element in P^ and p9li°=f^iP7tf).In general, for wî<j3 let ^'m be the first element in the set F"-[PCM/^TO ], where H7 = {pY\ i <m}yj {pl''\ i <m}, and tir^fM").Let #*-{¿"'I * </3} u {/;'| * < j.}.Denote by ^,0 the first element in the set F"-[il_"MU/*(__^)], and by £¿°'°, the element /M(^'°).For l^i<|3 let p** and ^0i represent no elements.If the power of Z,, is <2N°, let p'f, where 7__7_S 10 and m<ß, represent no elements.For each i<ß, let