Abstract clones as noncommutative monoids I
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910799162769408 |
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| author | Bucciarelli, Antonio Curien, Pierre-Louis Salibra, Antonino |
| author_facet | Bucciarelli, Antonio Curien, Pierre-Louis Salibra, Antonino |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abstract clones as noncommutative monoids I Bucciarelli, Antonio Curien, Pierre-Louis Salibra, Antonino Category Theory 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. |
| title | Abstract clones as noncommutative monoids I |
| topic | Category Theory |
| url | https://arxiv.org/abs/2501.14799 |