Endoprimal algebras

Abstract.

An algebra A is endoprimal if, for all $k \in \Bbb N$ the only maps from A ^k to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras, semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that this fails in general by proving that ${\bf 2}^2 \oplus {\bf 1}$, regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable.