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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2304.12807 |
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| _version_ | 1866917595843657728 |
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| author | Vucaj, Albert Zhuk, Dmitriy |
| author_facet | Vucaj, Albert Zhuk, Dmitriy |
| contents | We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form $f(x_1,\dots,x_n)\approx g(y_1,\dots,y_m)$, also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the $\operatorname{CSP}$ of a finite structure $\mathbb{A}$ only depends on the set of minor identities satisfied by the polymorphism clone of $\mathbb{A}$. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write $\mathcal{C} \preceq_{\mathrm{m}} \mathcal{D}$ if there exists a minor homomorphism from $\mathcal{C}$ to $\mathcal{D}$. We show that the aforementioned poset has only three submaximal elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12807 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Submaximal clones over a three-element set up to minor-equivalence Vucaj, Albert Zhuk, Dmitriy Rings and Algebras We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form $f(x_1,\dots,x_n)\approx g(y_1,\dots,y_m)$, also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the $\operatorname{CSP}$ of a finite structure $\mathbb{A}$ only depends on the set of minor identities satisfied by the polymorphism clone of $\mathbb{A}$. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write $\mathcal{C} \preceq_{\mathrm{m}} \mathcal{D}$ if there exists a minor homomorphism from $\mathcal{C}$ to $\mathcal{D}$. We show that the aforementioned poset has only three submaximal elements. |
| title | Submaximal clones over a three-element set up to minor-equivalence |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2304.12807 |