An algebraic theory of clones with an application to a question of Birkhoff and Maltsev

Abstract

We introduce the notion of clone algebra, intended to found a one-sorted, purely algebraic theory of clones. Clone algebras are defined by true identities and thus form a variety in the sense of universal algebra. We prove a representation theorem establishing the pertinence of our proposal. Moreover, we use clone algebras to answer a classical question about the lattices of equational theories, and we sketch some other applications.