Endoprimal implication algebras

Abstract.

An algebra A is endoprimal if, for all $n \in {\Bbb N}$, the only maps $f: A^n \rightarrow A$ which preserve the endomorphisms of A are the n-ary term functions of A. The theory of natural dualities has been a very effective tool for finding finite endoprimal algebras. We study endoprimality within the variety of implication algebras, which does not contain any non-trivial dualisable algebras. We show that there are no non-trivial finite endoprimal implication algebras. We also give some examples of infinite implication algebras which are endoprimal.