Absolute Extensors and Binary Monads

Abstract

Let F be a monad in the category Comp of compact Hausdorff spaces and continuous maps. An abstract convexity was constructed by Radul for each F-algebra of the monad F in the category Comp. It was proved that if the convexity of the monad F with some additional properties is binary then F has good topological properties, in particular, F X is an absolute extensor in the class of 0-dimensional spaces for each openly generated compactum X. We show in this paper that binarity is also a necessary condition.Keywords Monad · Convexity · Binarity · Absolute extensor in the class of 0-dimensional compacta Mathematics Subject Classification (2010)The notion of convexity considered in this paper is considerably broader than the classical one; specifically, it is not restricted to the context of linear spaces. Such convexities appeared in the process of studying different structures like partially ordered sets, semilattices, lattices, superextensions etc. We base our approach on the notion of topological convexity of [21] where the general convexity theory is covered from axioms to applications in different areas.T. Radul