On Equational Definability of Function Classes

We propose a notion of functional equation for functions of a fixed arity, which is based on a pair of clones. We present necessary conditions for a class of functions to be definable by such equations, and show that for certain choices of clones these conditions are also sufficient.

I. Basic notions and motivations

Throughout the paper A, B and are assumed to be finite sets. By a function of several variables from to (or simply function, when the sets and are clear from the context) we mean a map , where is called the arity of f. The set of all n-ary functions from to is denoted by . For a class , we set . Functions of several variables from the two-element set {a,1} to {a,1} are usually called Boolean functions. The set of all functions of several variables from to is denoted by .