Hemimetric-based λ-valued fuzzy rough sets

Abstract

A λ-subset, or a [0,λ]-valued fuzzy subset, is a mapping from a nonempty set to the interval [0,λ]. In this paper, we use the notion of hemimetrics, a kind of distance functions, as the basic structure to define and study fuzzy rough set model of λ-subsets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper/lower approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the related Lukasiewicz logical systems. We show that upper definable sets, lower definable sets and definable sets are equivalent, and they form an Alexandrov fuzzy topology. A processing of a λ-subset via fuzzy upper/lower approximation operators can actually considered as a processing of the related image, and thus has potential applications in image processing.