A general framework for product representations: bilattices and beyond

TL;DR

This paper introduces a general framework for representing algebraic structures like bilattices used in logic, providing a syntactic procedure for product construction and establishing categorical equivalences among varieties.

Contribution

It presents a universal product representation theorem with a syntactic duplication procedure, extending to a wide class of algebraic varieties and revealing categorical equivalences.

Findings

Established a categorical equivalence between base and enriched varieties.

Developed a syntactic duplication procedure for product algebra construction.

Applied the framework to various algebraic structures including trilattices.

Abstract

This paper studies algebras arising as algebraic semantics for logics used to model reasoning with incomplete or inconsistent information. In particular we study, in a uniform way, varieties of bilattices equipped with additional logic-related operations and their product representations. Our principal result is a very general product representation theorem. Specifically, we present a syntactic procedure (called duplication) for building a product algebra out of a given base algebra and a given set of terms. The procedure lifts functorially to the generated varieties and leads, under specified sufficient conditions, to a categorical equivalence between these varieties. When these conditions are satisfied, a very tight algebraic relationship exists between the base variety and the enriched variety. Moreover varieties arising as duplicates of a common base variety are automatically categorically equivalent to each other. Two further product representation constructions are also presented; these are in the same spirit as our main theorem and extend the scope of our analysis. Our catalogue of applications selects varieties for which product representations have previously been obtained one by one, or which are new. We also reveal that certain varieties arising from the modelling of quite different operations are categorically equivalent. Among the range of examples presented, we draw attention in particular to our systematic treatment of trilattices.