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.
Highlights
Our goal is to show that tools defined in individual theories, such as power set objects, various types of fuzzy type relations, variants of Zadeh’s extension or transformation operators defined using various types of fuzzy relations are only special examples of general constructions using monads in categories
This proves that all three theories, i.e., fuzzy soft sets, hesitant fuzzy sets and intuitionistic fuzzy sets, have a common background based on the theory of monads in categories
We develop the power set monad of L-fuzzy soft sets and we show that fuzzy soft relations are defined by this monad
Summary
There is a whole range of theories and theoretical structures that describe different types of non-probabilistic problems related to incomplete or inaccurate information. The huge number of publications in the form of monographs and papers dealing with both theory and applications of fuzzy sets testifies to their extraordinary importance from both a theoretical and especially a practical point of view. Inspired by the theory of fuzzy sets, a number of theories were created, which dealt with the issue of incomplete or inaccurate information and which had developed their own theory and a specific range of potential applications. The common feature of these three theories is, among other things, a large number of current publications dealing with both theoretical properties and application possibilities of these theories. For a basic overview of these theories, we will briefly mention some basic facts about these theories
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