Centralising groups of semiprojections and near unanimity operations

Abstract

Exploring the Galois correspondence induced by commutation between permutations and finitary operations, we study which permutation groups on finite sets arise as centralising groups of near unanimity operations and semiprojections, respectively. We derive upper and lower bounds for the Galois closures in a rather general setting and prove that, in fact, all permutation groups admit such a representation if the arity of the commuting functions is chosen high enough. Using concrete examples we also discuss limitations to such representations when the arity is (too) small.