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Abstract clones as noncommutative monoids I

Antonio Bucciarelli, Pierre-Louis Curien, Antonino Salibra

TL;DR

The paper develops a unifying algebraic framework for finitary and infinitary clones by introducing cm-monoids, merge algebras, and their related varieties. It proves a fundamental adjunction between clone algebras and cm-monoids, which restricts to an equivalence for finitely ranked / finite-dimensional cases, thereby linking abstract clones to finite-dimensional cm-monoids. It then introduces partial infinitary clone algebras (PICA) and shows an equivalence with extensional cm-monoids, extending the correspondence to infinitary settings. A companion program is outlined to study modules over cm-monoids and polymorphisms, hinting at applications to invariant relations and CSP complexity. Overall, the work provides a cohesive, category-theoretic bridge connecting clone theory, merge structures, and infinitary operations, enabling a unified treatment of polymorphisms and their algebraic properties.

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.

Abstract clones as noncommutative monoids I

TL;DR

The paper develops a unifying algebraic framework for finitary and infinitary clones by introducing cm-monoids, merge algebras, and their related varieties. It proves a fundamental adjunction between clone algebras and cm-monoids, which restricts to an equivalence for finitely ranked / finite-dimensional cases, thereby linking abstract clones to finite-dimensional cm-monoids. It then introduces partial infinitary clone algebras (PICA) and shows an equivalence with extensional cm-monoids, extending the correspondence to infinitary settings. A companion program is outlined to study modules over cm-monoids and polymorphisms, hinting at applications to invariant relations and CSP complexity. Overall, the work provides a cohesive, category-theoretic bridge connecting clone theory, merge structures, and infinitary operations, enabling a unified treatment of polymorphisms and their algebraic properties.

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.
Paper Structure (28 sections, 57 theorems, 51 equations)

This paper contains 28 sections, 57 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\omega$-Sequences
  4. Permutations
  5. Traces
  6. Operations and $\omega$-operations
  7. Clones of operations
  8. Abstract clones
  9. Clone algebras
  10. The variety of merge algebras
  11. The canonical merge algebras
  12. Some consequences of the axioms
  13. Restrictions
  14. Coordinates
  15. Degenerate merge algebras
  16. ...and 13 more sections

Key Result

Lemma 3.6

BS22 Let $\mathbf C$ be a clone algebra and $\mathbf b=b_0,\dots, b_{n-1}\in C$. If $k > n$ and $a\in C$ is independent of $\mathsf e_n,\mathsf e_{n+1}\dots,\mathsf e_{k-1}$, then

Theorems & Definitions (161)

  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Definition 3.5
  • Lemma 3.6
  • Remark 3.7
  • Remark 3.8
  • Lemma 3.9
  • proof
  • ...and 151 more