Submaximal clones over a three-element set up to minor-equivalence

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. 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 \({\text {CSP}}\) 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 \(\mathcal {C}\ {\preceq _{\textrm{m}}}\ \mathcal {D}\) if there exists a minor homomorphism from \(\mathcal {C}\) to \(\mathcal {D}\). We show that the aforementioned poset has only three submaximal elements.