**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.