Endoprimal distributive lattices are endodualisable

L. M/trki and R. P6schel have characterised the endoprimal distributive lattices as those which are not relatively complemented. The theory of natural dualities implies that any finite algebra A on which the endomorphisms of A yield a duality on the quasivariety DSP(A) is necessarily endoprimal. This note investigates endodualisability for finite distributive lattices, and shows, in a manner which elucidates Mfirki and P6schel's proof, that it is equivalent to endoprimality. Let (A; F) be an algebra and denote its endomorphism monoid by End A. Then A is said to be endoprimal if every finitary function on A commut ing with each element o f End A is a term function, and k-endoprimal (k >_ 1) if every function of arity not greater than k commut ing with all endomorphisms is a term function. In [15], L. Mfirki and R. P6schel proved that the following statements are equivalent for a non-trivial distributive lattice (L; v , A): (1) L is endoprimal, and (2) L is not relatively complemented. The contrapositive of the implication (1) ~ (2) is easy (by exploiting relative complementat ion, as a ternary function). As Mfirki and P6schel point out, their result shows that an endoprimal algebra need not generate a quasivariety with the kind of nice structural properties that are obtained when primality is extended in other ways, notably to quasiprimality (see [16]). Thus the occurrence and role of endoprimal algebras seems a little mysterious, and examples are rather scarce. However there is one situation in which endoprimal algebras arise very naturally, through duality theory. When we refer to a duality, we shall always mean a natural duality in the sense o f Davey and Werner. For our purposes, the recent survey [4] is the most convenient reference. A finite algebra A is said to be endodualisable if End A yields a duality on d = I~P(A) , in the sense defined in [4], Section 2. I f this Presented by A. F. Pixley. Received June 20, 1994; accepted in final form January 3, 1995. 1991 Mathematics Subject Classification. 06D05, 08B99.