Universal Clone Algebra

TL;DR

This paper introduces Universal Clone Algebra, a new algebraic framework unifying algebras and clones, and demonstrates its power by extending classical theorems like Birkhoff's HSP theorem.

Contribution

It develops a novel universal framework for algebras and clones, including the first introduction of t-algebras, and applies it to generalize key algebraic theorems.

Findings

Unified framework for algebras and clones

Concrete examples illustrating the framework's power

Extensions of Birkhoff's HSP theorem and topological variants

Abstract

We develop a new general framework for algebras and clones, called Universal Clone Algebra. Algebras and clones of finitary operations are to Universal Algebra what t-algebras and clone algebras are to Universal Clone Algebra. Clone algebras have been recently introduced to found a one-sorted, purely algebraic theory of clones, while t-algebras are first introduced in this article. We present a method to codify algebras and clones into t-algebras and clone algebras, respectively. We provide concrete examples showing that general results in Universal Clone Algebra, when translated in terms of algebras and clones, give more powerful versions of known theorems in Universal Algebra. We apply this methodology to Birkhoff's HSP theorem and to the recent topological versions of Birkhoff's theorem.