On the Semigroup of Equational Classes of Finite Functions

We consider the set of equtional classes of finite functions endowed with the operation of class composition. Thus defined, this set gains a semigroup structure. This paper is a contribution to the understanding of this semigroup. We present several interesting properties of this semigroup. In particular, we show that it constitutes a topological semigroup that is profinite.

I. Introduction

Throughout this paper, let be a finite nonempty set. Without loss of generality, we assume that for some natural number . An -ary function on is a mapping . By a class (of functions) on we simply mean a set of such mappings of possibly different arities. In this paper we shall be particularly interested in classes of functions definable by (depth-l) functional equations. In [6] it was shown that such classes, which we refer to as being equational, are exactly those classes that are closed under identifications and permutations of variables as well as addition and deletion of inessential variables. For further background and variants, see e.g. [4], [6], [8], [9], [14].