A Duality for Quasi Ordered Structures (i)

Recently, several authors extended Priestley duality for distributive lattices [9] to other classes of algebras, such as, e.g. distributive lattices with operators [7], MV -algebras [8], MTL and IMTL algebras [2]. In [5] necessary and sufficient conditions for a normally presented variety to be naturally dualizable, in the sense of [6], i.e. with respect to a discrete topology, have been provided. Under this perspective, also bounded distributive quasi lattices (bdq-lattices), introduced in [4], are naturally dualizable. Nonetheless, quasi lattices, constitute a generalization of lattice ordered structures to quasi ordered ones, i.e. structures in which the ordering relation is reflexive and transitive, but it may fail to be antisymmetric. Consequently, a sensible question arises: is there any “natural” candidate which stands to Priestley spaces as bounded distributive quasi lattices stand to bounded distributive lattices? In this talk, following an idea from [1], where a representation of bdq-lattices has been proposed, we present an alternative form of dualization of bdqlattices via the notion of preordered Priestley spaces. Preordered Priestley spaces are, in our opinion, of interest in that they naturally generalize Priestley spaces to a preordered setting, and, at the same time, they share with Priestley spaces desirable features. We will see that preordered Priestley spaces interpret, with respect to bdq-lattices, exactly the same rôle Priestley spaces play with respect to bounded distributive lattices.