From clones to cm-monoids

Abstract

Clones of functions play a foundational role in both universal algebra and theoretical computer science. In this work, we introduce clone merge monoids (cm-monoids), a unifying one-sorted algebraic framework that integrates abstract clones, clone algebras (previously introduced by the first and the third author), and Neumann's aleph0-abstract clones, while modelling the interplay of infinitary operations. Cm-monoids combine a monoid structure with a new algebraic structure called merge algebra, capturing essential properties of infinite sequences of operations. We establish a categorical equivalence between clone algebras and finitely-ranked cm-monoids. This equivalence yields by restriction a three-fold equivalence between abstract clones, finite-dimensional clone algebras, and finite-dimensional, finitely ranked cm-monoids, and is itself obtained by restriction from a categorical equivalence between partial infinitary clone algebras (which generalise clone algebras) and extensional cm-monoids. In a companion work, we develop the theory of modules over cm-monoids, offering a unified approach to polymorphisms and invariant relations, in the hope of providing new insights into algebraic structures and CSP complexity theory.