On Unification of Methods in Theories of Fuzzy Sets, Hesitant Fuzzy Set, Fuzzy Soft Sets and Intuitionistic Fuzzy Sets

Abstract

The main goal of this publication is to show that the basic constructions in the theories of fuzzy sets, fuzzy soft sets, fuzzy hesitant sets or intuitionistic fuzzy sets have a common background, based on the theory of monads in categories. It is proven that ad hoc defined basic concepts in individual theories, such as concepts of power set structures in these theories, relations or approximation operators defined by these relations are only special examples of applications of the monad theory in categories. This makes it possible, on the one hand, to unify basic constructions in all these theories and, on the other hand, to verify the legitimacy of ad hoc definitions of these constructions in individual theories. This common background also makes it possible to transform these basic concepts from one theory to another.

Cross-Border E-Commerce Supply Chain Risk Evaluation with FUZZY-ISM Model

“…Step 2: establish a fuzzy adjacency matrix Step 3: use the Chad fuzzy operator in the fuzzy matrix [39] calculation rules to calculate the fuzzy reachability matrix from the fuzzy adjacency matrix Step 4: introduce the intercept coefficient α to transform it into a reachable matrix of 0 to 1, and obtain the α intercept matrix…”

Fuzzy Transforms for Hesitant, Soft or Intuitionistic Fuzzy Sets

“…In this section, we deal with the construction of the intuitionistic F-transform (i.e., F-transform for intuitionistic L -fuzzy sets) using a monad and semimodule homomorphisms. For this purpose, we use the monad defined by the power set structure of all intuitionistic L -fuzzy sets in a set X, the existence of which was proven in the [39] and which is defined in the following definition.…”

“…Definition 13 [39] Let L be a complete MV-algebra. The monad = (J, ⊠, 𝜚) in the category is defined by…”

“…In our previous paper [39], we tried to unify some of methods used in all these theories. For this purpose, we used a special tool from the theory of categories, namely the theory of monads (see, e.g., [35,38,47,48]), which allows to unify not only various types of relations but also transformation operators defined by these relations.…”

“…For this purpose, we used a special tool from the theory of categories, namely the theory of monads (see, e.g., [35,38,47,48]), which allows to unify not only various types of relations but also transformation operators defined by these relations. In the present paper, we use these results concerning monads from [39] and we focus on the F-transform method and its possible modifications in these structures. For such F-transform variants, we will require that their definitions are based on the same principles as the F-transforms for classical L-fuzzy sets, where L is a complete residuated lattice.…”