References

In this work we develop a categorical duality for certain classes of many-sorted algebras, called many-sorted lattices because each sort admits a structure of distributive lattice. This duality is strongly based on the Priestley duality for distributive lattices developed in [3] and [4] and on the representation of many sorted lattices with operators given by Sofronie-Stokkermans in [6]. In this last paper the author describes a way of represent a many sorted lattice with dier-ent operator by means of a family of Priestley spaces with additional relations. In this paper we will formally complete the duality between these structures, by establishing the arrows in each category and proving the dual equivalence between them. This duality applied to the single sort case, that is the case of distributive lattices with operators coincide with the duality developed in [5] by Soronie Stokermans, and generalize many other dualities and representa-tions. We will use the duality for the case of distributive lattice with operators to describe the congruences, simple and subdirectly irreducible algebras and subalgebras. These results include, in its applications to some particular cases, the ones obtained in [2] for J-Lattices, [7] for Ockham algebras, and [1] for distributive lattices with fusion and implication, to name some of them