On the categoricity of semigroup-theoretical properties

Abstract

By asemigroup-theoretical property we mean a property of semigroups which is preserved by isomorphism. Such a property iscategorical if it can be expressed in the language of categories: roughly, without using elements. We show that this is always possible with the proviso that in the case of one-sided properties we cannot refer in categorical terms to a specific side. For example, the property of having aleft identity cannot be described categorically in the category of semigroups, since the functor ()^op which takes a semigroup into its “opposite” semigroup is a category automorphism. We show that ()^op is the only non-trivial automorphism of the category of semigroups (up to natural equivalence of functors). In other words, the “automorphism group” of the category of semigroups has order two.