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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