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.Comment: 20 pages 2 table
