Dualities and Identities

H. A. Priestley

There is a dual category equivalence between distributive lattices with 0,1 and the category, P, of Priestley spaces (X; T, <). On top of this duality ride numerous dualities for varieties of distributive lattices with additional operations, many of them of interest in algebraic logic and universal algebra: de Morgan algebras, Stone algebras, Heyting algebras, l-groups, MV-algebras, ... . Typically, a variety A of algebras (A;\land,\lor,0,1, {f_\mu}) is dual to a class of structures of the form (X; T, {<}\cup R), where (X; T, <) is in P and (X; {<}\cup R) is a relational structure compatible with the topology in an appropriate sense. We survey the general features of this translation process, and indicate how in certain cases it is possible algorithmically to determine the identities (of a suitable canonical type) satisfied by a finite algebra in A by writing down sentences satisfied in the dual relational structure.