Natural dualities through product representations: bilattices and beyond

TL;DR

This paper develops a systematic method to construct natural dualities for bilattice-based algebraic varieties using product representations, enabling new dualities for many such varieties, including those with generalized conflation.

Contribution

It introduces a unified algebraic framework combining product representations with natural duality theory to generate dualities for bilattice varieties, many of which were previously unknown.

Findings

Constructed natural dualities for a wide class of bilattice varieties.

Developed dualities for bilattices with generalized conflation operations.

Outlined procedures for pre-bilattice-based algebras without negation.

Abstract

This paper focuses on natural dualities for varieties of bilattice-based algebras.Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation (not assumed to be an involution or commute with negation). Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras (so negation is absent).