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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.04034 |
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| _version_ | 1866912874866147328 |
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| author | Fioravanti, Stefano Kompatscher, Michael Rossi, Bernardo |
| author_facet | Fioravanti, Stefano Kompatscher, Michael Rossi, Bernardo |
| contents | Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\mathbf A$ and $\mathbf B$ there are only finitely many clonoids from $\mathbf A$ to $\mathbf B$ if and only if $\mathbf A$, $\mathbf B$ are of coprime order.
We confirm this conjecture for a broad class of modules $\mathbf A$. In particular we show that, if $\mathbf A$ is a finite $k$-dimensional vector space, then every clonoid from $\mathbf A$ to a coprime module $\mathbf B$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\mathbf A,\mathbf B)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04034 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Clonoids over vector spaces Fioravanti, Stefano Kompatscher, Michael Rossi, Bernardo Rings and Algebras Discrete Mathematics 08A68, 08A70, 08A40 Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\mathbf A$ and $\mathbf B$ there are only finitely many clonoids from $\mathbf A$ to $\mathbf B$ if and only if $\mathbf A$, $\mathbf B$ are of coprime order. We confirm this conjecture for a broad class of modules $\mathbf A$. In particular we show that, if $\mathbf A$ is a finite $k$-dimensional vector space, then every clonoid from $\mathbf A$ to a coprime module $\mathbf B$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\mathbf A,\mathbf B)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time. |
| title | Clonoids over vector spaces |
| topic | Rings and Algebras Discrete Mathematics 08A68, 08A70, 08A40 |
| url | https://arxiv.org/abs/2602.04034 |