Binary central relations and submaximal clones determined by nontrivial equivalence relations

Abstract

Let k ≥ 3, θ a nontrivial equivalence relation on E k = {0, . . . ,k – 1}, and ρ a binary central relation on E k (a reflexive graph with a vertex having E k as its neighborhood). It is known that the clones Pol θ and Pol ρ (of operations on E k preserving θ and ρ, respectively) are maximal clones; i.e., covered by the largest clone in the inclusion-ordered lattice of clones on E k . In this paper, we give the classification of all binary central relations ρ on E k such that the clone Pol θ ∩ Pol ρ is maximal in Pol θ.