Dualisability in general and endodualisability in particular

Abstract

Sufficient conditions are given for the dualisability of a finite algebra M in the quasivariety generated by a dualisable algebra D. The dual structure on M ∼ may be obtained by adding a suitable subset of EndM to the structure inherited from D ∼ . The results show that every finite abelian group is dualisable and that as dual structure we may take the original group operations along with the endomorphism monoid. Similar results follow for finite distributive lattices, Boolean algebras, semilattices and vector spaces, for example. The results are applied to yield large classes of endodualisable algebras. For example, it is proved that a finite vector space over a finite field is endodualisable if and only if it doesn’t have dimension 1, that every finite boolean algebra and every finite algebra in the class of Heyting algebras generated by a threeelement chain is endodualisable, while a finite Stone algebra is endodualisable if and only if it is boolean or its dense set is nonboolean. We also obtain the result of Davey, Haviar and Priestley that a distributive lattice is endodualisable if and only if it is nonboolean. Endodualisable algebras are of interest because they are necessarily endoprimal, that is, their term functions are just the finitary maps which preserve the action of the endomorphism monoid.