Equivalence of operations with respect to discriminator clones

For each clone CC on a set AA there is an associated equivalence relation, called CC-equivalence, on the set of all operations on AA, which relates two operations iff each one is a substitution instance of the other using operations from CC. In this paper we prove that if CC is a discriminator clone on a finite set, then there are only finitely many CC-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.