Priestley Duality for Many Sorted Algebras and Applications
نویسندگان
چکیده
In this work we develop a categorical duality for certain classes of manysorted 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 di erent 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 So ronie Stokermans, and generalize many other dualities and representations. 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.
منابع مشابه
Dualities in Lattice Theory
In this note we prove several duality theorems in lattice theory. We also discuss the connection between spectral spaces and Priestley spaces, and interpret Priestley duality in terms of spectral spaces. The organization of this note is as follows. In the first section we collect appropriate definitions and basic results common to many of the various topics. The next four sections consider Birk...
متن کاملPriestley Duality for SHn-Algebras and Applications to the Study of Kripke-Style Models for SHn-Logics
متن کامل
A non-commutative Priestley duality
We prove that the category of left-handed skew distributive lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a noncommutative version of classical Priestley duality for distributive lattices. The result also generalizes the recent development of Stone duality for skew Boolean algebras.
متن کاملOptimal natural dualities for varieties of Heyting algebras
The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality...
متن کاملPriestley duality for N4-lattices
We present a new Priestley-style topological duality for bounded N4-lattices, which are the algebraic counterpart of paraconsistent Nelson logic. Our duality differs from the existing one, due to Odintsov, in that we only rely on Esakia duality for Heyting algebras and not on the duality for De Morgan algebras of Cornish and Fowler. A major advantage of our approach is that for our topological ...
متن کامل