Multifunctions and the fixed point property for products of ordered sets
\textit{J. M. Walker} [Discrete Math. 48, 275-288 (1984; Zbl 0546.06003)] attached to every ordered set (1) (E,\(\leq)\) a quasi-order (2) \((P(E,\leq),\leq_ W)\) of the power set PE of E be defining \(A\leq_ WB\) iff \(A\subset ideal B \& B\subset filter A\), for any A,B\(\subset E\). The present author says that a multifunction F: (1)\(\to (M,\leq_ M)\) \((:=any\) ordered set) is order-preserving or increasing iff \(x\leq y\) implies \(Fx\leq_ WFy\) [cf. \textit{R. E. Smithson}, Proc. Am. Math. Soc. 28, 304-310 (1971; Zbl 0238.06003)] and (1) has GFPP \((:=general\) fixed point property) iff (2) has FPP. The main aim of the author was to improve Theorem 7.4 of Walker, loc. cit., by removing the property of chain completeness of factors and proving Theorem 1: If one of the ordered sets A,B has GFPP and the other one has FPP, then their product \(A\times B\) has FPP; Theorem 2: (1) is chain complete and for every increasing multifunction F: (1)\(\to (M,\leq_ M)\) there are comparable points \(e_ 0,e_ 1\in (1)\) such that \(e_ 1\in Fe_ 0\), then FPP of \((M,\leq_ M)\) implies the FPP of the product \(E\times M\).
scientific article; zbMATH DE number 3948342