Commutation for Functions of Small Arity Over a Finite Set

Abstract

Commutation is defined for multi-variable functions on a finite base set. For a set F of functions the centralizer F* of F is the set of functions which commute with all functions in F. For a function f a minor of f is a function obtained from f by iden- tifying some of its variables. An important observation is that the centralizer f* of f is a subclone of the centralizer of any minor of f, which motivates the study of the centralizers of functions of small arity. In this paper we determine the centralizers of all 2-variable functions over the two-element set. Then, as a generalization of AND on the 2-element set we consider the function Min on the k-element set, k > 1, and characterize the centralizer of Min using a term from lattice theory.