Abstract clones as noncommutative monoids I

TL;DR

This paper introduces clone merge monoids (cm-monoids), a new algebraic framework unifying various clone concepts and modeling infinite operations, establishing categorical equivalences that deepen the understanding of algebraic structures in universal algebra and computer science.

Contribution

The paper defines cm-monoids, proves their categorical equivalence with clone algebras, and unifies multiple clone concepts into a single algebraic framework, extending the theory of clones.

Findings

Established categorical equivalence between clone algebras and finitely-ranked cm-monoids.

Unified abstract clones, clone algebras, and Neumann's aleph0-abstract clones within a single framework.

Provided a foundation for future work on modules over cm-monoids and applications to CSP complexity.

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.