Residuated lattices arising from equivalence relations on Boolean and Brouwerian algebras

Abstract

AbstractLogics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper.Namely, we consider pairs consisting of a Brouwerian algebra (i. e. a dual Heyting algebra) and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s‐equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL‐algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV‐algebras. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)