An algebraic theory of clones

Abstract

We introduce the notion of clone algebra ($\mathsf {CA}$), intended to found a one-sorted, purely algebraic theory of clones. $\mathsf {CA}$s are defined by identities and thus form a variety in the sense of universal algebra. The most natural $\mathsf {CA}$s, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras ($\mathsf {FCA}$). The universe of a $\mathsf {FCA}$, called $\omega$-clone, is a set of infinitary operations on a given set, containing the projections and closed under finitary compositions. The main result of this paper is the general representation theorem, where it is shown that every $\mathsf {CA}$ is isomorphic to a $\mathsf {FCA}$ and that the variety $\mathsf {CA}$ is generated by the class of finite-dimensional $\mathsf {CA}$s. This implies that every $\omega$-clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We conclude the paper with two applications. In the first one, we use clone algebras to give an answer to a classical question about the lattices of equational theories. The second application is to the study of the category of all varieties of algebras.