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

TL;DR

This paper develops a new algebraic framework called clone algebras to represent and analyze clones of functions, providing a universal algebraic perspective and applications to lattice theory and varieties.

Contribution

It introduces clone algebras as a purely algebraic theory of clones, proves a representation theorem, and connects clone algebras to the category of varieties, offering new insights and tools.

Findings

Every clone algebra is isomorphic to a functional clone algebra.

The variety of clone algebras is generated by block algebras.

The category of varieties is categorically isomorphic to a subcategory of clone algebras.

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. The most natural clone algebras, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras. The universe of a functional clone algebra, called omega-clone, is a set of infinitary operations containing the projections and closed under finitary compositions. We show that there exists a bijective correspondence between clones (of finitary operations) and a suitable subclass of functional clone algebras, called block algebras. Given a clone, the corresponding block algebra is obtained by extending the operations of the clone by countably many dummy arguments. One of the main results of this paper is the general representation theorem, where it is shown that every clone algebra is isomorphic to a functional clone algebra. In another result of the paper we prove that the variety of clone algebras is generated by the class of block algebras. 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 answer a classical question about the lattices of equational theories. The second application is to the study of the category VAR of all varieties. We introduce the category CA of all clone algebras (of arbitrary similarity type) with pure homomorphisms as arrows. We show that the category VAR is categorically isomorphic to a full subcategory of CA. We use this result to provide a generalisation of a classical theorem on independent varieties.