Priestley duality for bilattices
A pre-bilattice is a set equipped with two separate lattice structures. Such algebras have been used as the basis for a denotational semantics for systems of inference that arise in artificial intelligence and knowledge-based logic programming. A pre-bilattice is distributive if all possible distributive laws hold. A bilattice is a pre-bilattice equipped with an additional unary involution (negation) which reverses both induced order relations. Every (distributive) pre-bilattice can be represented as a special direct product of two (distributive) lattices. Every (distributive) bilattice can be represented as a special direct product of a (distributive) lattice with itself. In fact, the category of bounded distributive pre-bilattices is equivalent to the product category of bounded distributive lattices with itself. Similarly, the category of bounded distributive bilattices is equivalent to the category of bounded distributive lattices; see [\textit{B. Mobasher} et al., Algebra Univers. 43, No. 2--3, 109--125 (2000; Zbl 1012.06008)]. It is therefore possible to use Priestley duality for the representation of bounded distributive pre-bilattices and bilattices. In the present paper, the authors give details of this representation. Next, they consider pre-bilattices and bilattices with additional operations and show how to adapt Priestley duality to obtain dualities for bilattices with conflation and Brouwerian bilattices.
scientific article; zbMATH DE number 6088859
scientific article