Equivalence of operations with respect to discriminator clones

TL;DR

This paper investigates the structure of operation equivalences induced by discriminator clones on finite sets, showing finiteness of classes and providing explicit descriptions for Boolean functions.

Contribution

It proves finiteness of C-equivalence classes for discriminator clones and characterizes these classes explicitly for Boolean functions.

Findings

Finitely many C-equivalence classes for discriminator clones on finite sets.

The smallest discriminator clone is minimal regarding the finiteness property.

Explicit descriptions of equivalence relations for Boolean discriminator clones.

Abstract

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.