A duality theory for bilattices
Bilattices are algebras with two separate lattice structures. They have been used as the basis for a denotational semantics for systems of inference that arise in artificial intelligence and knowledge-based programming. Recent studies of the algebraic properties of bilattices have provided insight into their internal structures, and have led to practical results, especially in reducing the computational complexity of bilattice-based multivalued logic programs. In the paper the representation problem for interlaced bilattices without negation raised by \textit{A. Romanowska} and \textit{A. Trakul} [Universal and Applied Algebra, Proc. 5th Symp. Turawa/Pol. 1988, 235-253 (1989; Zbl 0738.06008)] and extended to arbitrary interlaced bilattices without negation by \textit{A. Avron} [Math. Struct. Comput. Sci. 6, 287-299 (1996; Zbl 0856.06005)] is presented. A natural equivalence is then established between the category of interlaced bilattices and the cartesian square of the category of bounded lattices. As a consequence a dual natural equivalence is obtained between the category of distributive bilattices and the coproduct of the category of bounded Priestley spaces with itself. Some applications of these equivalences are given. The subdirectly irreducible interlaced bisemilattices are characterized in terms of subdirectly irreducible lattices.
scientific article; zbMATH DE number 1899929