In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these operators (...) are presented. (shrink)

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes.

The lack of double negation and de Morgan properties makes fuzzy logic unsymmetrical. This is the reason why fuzzy versions of notions like closure operator or Galois connection deserve attention for both antiotone and isotone cases, these two cases not being dual. This paper offers them attention, comming to the following conclusions: – some kind of hardly describable ‘‘local preduality’’ still makes possible important parallel results; – interesting new concepts besides antitone and isotone ones (like, for instance, conjugated pair), that (...) were classically reducible to the first, gain independency in fuzzy setting. (shrink)

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.