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Extension theorems in axiomatic theory of convexity. (English) Zbl 0954.52001
A convexity space is a set \(X\) together with a collection of subsets, called convex sets, which contains the empty set and is closed under intersection of arbitrary subcollections and under union of chains. Given two convexity spaces \(X\) and \(Y\), a map \(f:X\to Y\) is said to be convexity preserving if \(f^{-1}(G)\) is convex in \(X\) for each convex set \(G\) in \(Y\).
The paper deals with conditions for a map \(f:M\subset X\to Y\) to have a convexity preserving extension \(g:X\to Y\). In the particular case when the convexity spaces are generated by lattice structures the author obtains a condition for the existence of an homomorphism \(g:K\to L\) between lattices extending a map \(f:M\subset K\to L\), which generalizes the Extension Criterion of R. Sikorski [‘Boolean algebras’, Springer-Verlag (1960; Zbl 0087.02503)] for Boolean algebras. Special attention is paid to extension properties of maps defined on \(S_4\) convexity spaces (that is, convexity spaces such that all points are convex and any two disjoint convex sets are contained in complementary convex sets). In particular, an analogue of Tietze-Urysohn’s Extension Theorem for maps from subsets of \(S_4\) convexity spaces into complete Boolean algebras is obtained.
MSC:
52A01 Axiomatic and generalized convexity
06B99 Lattices
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