Implicit operations on certain classes of semigroups

Abstract

Varieties of algebras are characterized by identities, where an identity is a formal equality of two terms (i.e., operations defined by means of the underlying operations). Analogously, pseudovarieties of (finite) algebras are defined by pseudo-identities, these being formal equalities of so-called implicit operations (briefly, functions compatible with all homomorphisms). To further explore this analogy to yield results on finite algebras, it is necessary to obtain clear descriptions of implicit operations. This work is a contribution to this project in the area of semigroup theory. All unary implicit operations on semigroups are described, and the implicit operations on certain pseudovarieties of semigroups are given in terms of "generating" operations. The existence of some unusual implicit operations is established based on classical combinatorial theorems about words.