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

Abstract

AbstractWe 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 $$f(x_1,\dots ,x_n)\approx g(y_1,\dots ,y_m)$$ f ( x 1 , ⋯ , x n ) ≈ g ( y 1 , ⋯ , y m ) , also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the $${\text {CSP}}$$ CSP of a finite structure $$\mathbb {A}$$ A only depends on the set of minor identities satisfied by the polymorphism clone of $$\mathbb {A}$$ 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 $$\mathcal {C}\ {\preceq _{\textrm{m}}}\ \mathcal {D}$$ C ⪯ m D if there exists a minor homomorphism from $$\mathcal {C}$$ C to $$\mathcal {D}$$ D . We show that the aforementioned poset has only three submaximal elements.