Endoprimal Monoids and Witness Lemma in Clone Theory

For a fixed set $A$, an endoprimal monoid $M$ is a set of unary functions on $A$ which commute with some set $F$ of functions on $A$. A member of such $M$ defines an endomorphism on $F$. It is known to be hard to effectively characterize such endoprimal monoids. In this paper we present and discuss the ''witness lemma'' to study endoprimal monoids. Then, for the case where $|A|=3$, we verify two monoids to be endoprimal and then determine all endoprimal monoids having subsets of unary functions as their witnesses.