Characterization of sets of discrete functions by algebraic identities

Abstract

In [3] the authors address the question of which classes of Boolean functions can be characterized by algebraic identities. This question is answered for classes of Boolean functions closed under permutation of variables and addition or deletion of fictitious variables. Almost all classes of Boolean functions have these properties. In this paper we generalize this question to classes of operations defined on an arbitrary finite set A with |A|≥2 and using the universal-algebraic concept of an identity as a pair of terms satisfied in a given algebra. We introduce a Galois connection between classes of n-ary operations defined on the same finite set and sets of equations of type τ=(n). Moreover, we define certain operators on classes of operations which preserve the Galois-closed sets of operations with respect to this Galois connection. Finally we prove that the Galois-closed sets of operations are unions of n-ary clones iff the defining identities are satisfied as hyperidentities.