Duality Theory: Hybrids

Abstract

This chapter covers the duality theory for closure algebras and Heyting algebras. The notion of a hybrid of topology and order is introduced, and the fundamental properties of hybrids are studied. It is proved that the category of hybrids and hybrid maps is dually equivalent to the category of closure algebras and closure algebra homomorphisms, while the category of strict hybrids and hybrid maps is dually equivalent to the category of Heyting algebras and Heyting homomorphisms. The notion of a Grzegorczyk algebra is introduced, and several characterizations of these algebras are given using the duality theory. From these characterizations it follows that the category of skeletal closure algebras is a subcategory of the category of Grzegorczyk algebras. Finally, it is proved that the variety of Grzegorczyk algebras is generated by its finite members, and some consequences of this result are derived.