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Maximal elements and fixed points for binary relations on topological ordered spaces. (English) Zbl 0852.90006

Summary: Topological semilattices are partially ordered topological spaces \(X\) in which each pair of elements \(x\), \(x'\in X\) has a least upper bound \(x\vee x'\) and the function \((x, x')\mapsto x\vee x'\) is continuous. We establish in such a context an order theoretical version of the classical result of Knaster-Kuratowski-Mazurkiewicz, as well as fixed point theorems for multivalued mappings. One can then, as in the context of topological vector spaces, obtain existence results for the largest elements of a weak preference relation or maximal elements for a strict preference relation. Beyond these particular results, we wish to attract attention to path-connected topological semilattices, examples of which will be found in the introduction, and their rich geometric structure – a geometric structure rich enough to provide order theoretical versions of some of the basic tools from mathematical economics and, therefore, also an alternative to the usual convexity assumptions.

MSC:

91B08 Individual preferences
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