Rough Semiring-Valued Fuzzy Sets with Application

Abstract

Many of the new fuzzy structures with complete MV-algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called AMV-algebras. The category of complete AMV-algebras is isomorphic to the category of special pairs (R,R*) of complete commutative semirings and the corresponding fuzzy sets are called (R,R*)-fuzzy sets. We use this theory to define (R,R*)-fuzzy relations, lower and upper approximations of (R,R*)-fuzzy sets by (R,R*)-relations, and rough (R,R*)-fuzzy sets, and we show that these notions can be universally applied to any fuzzy type structure that is transformable to (R,R*)-fuzzy sets. As an example, we also show how this general theory can be used to determine the upper and lower approximations of a color segment corresponding to a particular color.

Cut Systems with Relational Morphisms for Semiring-Valued Fuzzy Structures

“…The basic value structures that we use in the paper are dual pairs of semirings as the equivalent form of AMV-algebras. All these notions have been introduced in a recent paper [7], and it is, therefore, appropriate to repeat the basic definitions and properties of these new structures. In this section, we present the basic definitions and properties of these structures.…”

“…Since the reverse transformation of the results related to the theory of (R, R * )-fuzzy sets to the results related to the new fuzzy structure is relatively simple, it is advisable to develop the theory of (R, R * )-fuzzy sets as much as possible so that it is possible to subsequently transform these results on analogical theory in new fuzzy structures. In previous papers, we dealt with, e.g., definitions and properties of the theory of approximations of (R, R * )-fuzzy sets, as well as definitions and properties of rough (R, R * )-fuzzy sets [7] or F-transform theory for (R, R * )-fuzzy sets [6]. All these notions can be relatively simply transformed (without any additional proofs) into analogical notions with analogical properties into new fuzzy structures that can be transformed into (R, R * )-fuzzy sets; moreover, due to the existence of two monads, the results are mostly defined in two adjoint variants.…”