VARIETIES OF DISTRIBUTIVE LATTICES WITH UNARY OPERATIONS .1.

A unified study is undertaken of finitely generated varieties HSP((P) under bar) of distributive lattices with unary operations, extending w ork of Cornish. The generating algebra (P) under bar is assumed to be of the form (P; boolean AND, boolean OR, 0, 1, {f(mu)}), where each f( mu) is an endomorphism or dual endomorphism of(P; boolean AND, boolean OR, 0, 1), and the Priestley dual of this lattice is an ordered semig roup N whose elements act by left multiplication to give the maps dual to the operations f(mu). Duality theory is fully developed within thi s framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for WS((P) under bar) to be essentially the same, so that, inter alia, co products in HSP((P) under bar) are enriched D-coproducts.