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

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(x₁, ,xₙ)≈ g(y₁, ,yₘ) f(x1,⋯,xn)≈g(y1,⋯,ym) , also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the CSP CSP of a finite structure A A only depends on the set of minor identities satisfied by the polymorphism clone of 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 C\ ₘ\ D C⪯mD if there exists a minor homomorphism from C C to D D . We show that the aforementioned poset has only three submaximal elements.