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A characterization of complete Boolean algebras

Kubiś W.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 1

91-95

Every lattice and, in particular, every Boolean algebra is a convexity space with a naturally defined convexity structure. We characterize complete Boolean algebras as the only S3 convexity spaces having an extension property for certain classes of convexity preserving maps. This answers our question posed in [1]. Our characterization provides also a short proof of Sikorski's extension theorem for homomorphisms of Boolean algebras.

Frechet type theorem and its applications to multifunctions

Kubiś B., Kubiś W.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

165-170

We state a Frechet type theorem for measurable maps with values in an almost arcwise connected metrizable space. As an application, we obtain some results on continuous approximation of measurable multifunctions.

Extension theorems in axiomatic theory of convexity

Kubiś W.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

89-96

We present a criterion for extending convexity preserving maps of convexity spaces. In a special case of convexity generated by a lattice structure this gives Sikorski's Extension Criterion for extending of maps of lattices. We also consider the class of convexity absolute extensors. It appears that complete Boolean algebras with a natural convexity belong to this class. In particular, we present an analogue of Tietze-Urysohn's Extension Theorem for maps of convexity spaces with values in a complete Boolean algebra.