Cut approach to islands in rectangular fuzzy relations

Abstract

The paper investigates fuzzy relations on a finite domain in the cutworthy framework, dealing with a new property coming from the information theory. If the domain of a relation is considered to be a table, then a rectangular subset of the domain whose values under this relation are greater than the values of all neighboring fields is called an island. Consequently, the so called rectangular fuzzy relations are introduced; their cuts consist of rectangles as sub-relations of the corresponding characteristic functions. A characterization theorem for rectangular fuzzy relations is proved. We also prove that for every fuzzy relation on a finite domain, there is a rectangular fuzzy relation with the same islands, and an algorithm for a construction of such fuzzy relations is presented. In addition, using methods developed for fuzzy structures and their cuts, we prove that for every fuzzy relation there is a lattice and a lattice valued relation whose cuts are precisely the islands of this relation. A connection of the notion of an island with formal concept analysis is presented.