Fixed Points of Products and Ordered Sums of Simply Ordered Sets
نویسنده
چکیده
Let A and B be simply ordered sets and A XB the cartesian product ordered by first differences. Sufficiency conditions are given on the sets A and B for the existence of fixed points in the set AXB (Theorem 2). Sufficiency conditions are given on the sets A, B, and C for the incomparability of the order types1 \A XB\ and \A XC\ (| CXA\ and | CXB\) (Corollary 2 of Theorem 7). The content of Theorem 4 is that if {A^} is a family of 2^0 linear sets, each containing a fixed point, then there exists a subset G of RXR, where R denotes the real numbers, such that for each element x in R, if Bx = {y\ (x, y) £G}, either Bx is empty or is one of the sets A%. Furthermore, if p is a fixed point in Bx, then (x, p) is a fixed point in G.
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