A Note on Witnesses of Centralizing Monoids (Algebraic system, Logic, Language and Computer Science)

Abstract

We consider multi-variable functions defined over a fixed finite set A. A centralizing monoid M is a set of unary functions on A which commute with all members of some set F of functions on A, where F is called a witness of M. We show that every centralizing monoid has a witness whose arity does not exceed |A|. Then we present a method to count the number of centralizing monoids which have sets of some specific functions as their witnesses. Finally, some results on the three-element set E3 are reported concerning witnesses consisting of binary idempotent functions, majority functions or ternary semiprojections.