Dual categories for endodualisable Heyting algebras: optimization and axiomatization

Abstract

This paper is both a contribution to natural duality theory as it applies to varieties of Heyting algebras and to the theory of standard topological quasi-varieties (see [3]) in general. We prove that the n-element Heyting chain C n has an alter ego, consisting of n−2 endomorphisms, that yields an optimal duality on the variety generated by C n . In the case n = 4, we give a set of quasi-equations that describe the dual category. We also give a set of quasi-equations that describe the strong dual category that is obtained by adding a partial endomorphism of C4 to the type of the alter ego. A quasi-equational description of the optimal dual category for the variety of Heyting algebras generated by \({\bf 2}^{2} \oplus {\bf 1}\) is also given. En route, we prove that if a finite topological unary algebra is injective amongst the finite algebras in the variety it generates, then it is standard. The results of [3] then allow us to give non-topological proofs of our topological descriptions of the dual categories.