Representation of semigroups by products in a category

Abstract

Each finite cyclic group has a representation in the category of all Abelian groups (see [2]) and in the category of all Boolean algebras (see [3]). Each semigroup with one generator has a representation in the category of all modules over a suitable ring (see [l], [6]), and in the category of all topological spaces (see [9]). The method of [9] admits a large generalization. It is presented in this paper. We prove that each semigroup with one generator and each Abelian group have representations in each of the following categories: the category of all topological (or uniform or proximity) spaces, of all directed graphs, small categories, unary universal algebras with at least two operations and partial universal algebras of some types. Thus, taking the additive group of all rational numbers as a represented semigroup, we get a space (or a graph or an algebra) which has “negative powers” and “all roots.” It is a pleasure to acknowledge my indebtedness to M. Kat&ov for reading the manuscript and making valuable criticisms, and to my colleague A. Pultr for valuable suggestions.