Math. Log. Quart. zyxwvut 45 (1999) 2, 207 - 218 zyxwv Mathematical Logic Quarterly @ WILEY-VCH Verlag Berlin GmbH zy 1999 Distributive Lattices with a Negation Operator Sergio Arturo Celani') Departamento de Matembtica, Facultad de Ciencias Exactas Universidad Nacional del Centro, Pinto 399, 7000 Tandil, Argentina') Abstract. In this note we introduce and study algebras zyxwv (L, V, zyxw A, 1, 0,l) of type (2,2,1,1,1) such that zyxwvutsrq (L, V, A, 0,l) is a bounded distributive lattice and -,is an operator that satisfies the conditions -,(a V b) = -,a A -,b and -0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras. Mathematics Subject Classification: 06D05, 06D16, 03G10. Keywords: Bounded distributive lattice, Priestley space, Modal operator, Quasi-Stone algebra, De Morgan algebra, p-algebra. 1 Preliminaries In [6] it was shown that the intuitionistic negation can be treated as a modal impossi- bility operator. This approach can be generalized to a fragment {V, A, zyx 7, I, T} of in- tuitionistic logic or classical propositional logic. On the other hand, in [9] (see also [7]) the category of bounded distributive lattices with a modal operator that corresponds to a modal operator of possibility, was introduced. CIGNOLI, LAFALCE and PETRO- VICH [4] proved that this category is dual to the category of Priestley spaces with a Priestley relation. Now, we can ask, as in [6], what the space dual to a distributive lattice endowed with a negation operator is. In this note we shall consider the cate- gory of bounded distributive lattices endowed with a negation operator and we shall show that this category is dual to the category of Priestley spaces with a particular re- lation. In case that the negation operator is the operation of pseudocomplementation, we obtain the known duality between palgebras and pspaces (see [12]). The dual- ity reported in this work also generalizes the duality given in [8] for semi-DeMorgan algebras. In [13] the variety of quasi-Stone algebras was introduced as a generalization of the variety of Stone algebras. We shall see that this variety has many similar properties to the variety of bounded distributive lattices with a quantifier which was considered by R. CIGNOLI in [5]. would like to the thank the referee for the comments and suggestions on the paper. 'Ie-mail: scelani@exa.unicen.edu.ar