Submaximal clones over a threeelement set up to minorequivalence
Abstract
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. Minorequivalent 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 threeelement 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2023
 DOI:
 10.48550/arXiv.2304.12807
 arXiv:
 arXiv:2304.12807
 Bibcode:
 2023arXiv230412807V
 Keywords:

 Mathematics  Rings and Algebras