Closure theory for semirings-valued fuzzy sets with applications to new fuzzy structures

Abstract

Many of the new fuzzy structures, including intuitionistic, neutrosophic, or fuzzy soft sets, use their oven theories and methods for operations with their fuzzy structures, including topological constructions. In the paper we show how basic closure methods could be universally defined in a number of new fuzzy structures, without having to define a new theory for individual fuzzy structures. This approach is based on the transformation of these fuzzy structures into a new type of fuzzy set, called (R,R⁎)-fuzzy sets, whose value sets are special pairs (R,R⁎) of commutative and idempotent semirings. The main advantage of this procedure is that all theoretical results that are proved for (R,R⁎)-fuzzy sets can be relatively easily transformed into an analogous result in all new fuzzy structures that can be transformed into (R,R⁎)-fuzzy sets, without the need to prove this result for individual types of fuzzy structure.