Fuzzy Type Relations and Transformation Operators Defined by Monads

Abstract

Using the theory of monads in categories and the theory of monadic relations, the concept of general transformation operator defined by a monadic relation is introduced. It is proven that a number of standard relations used in categories of fuzzy structures are monadic relations for monads defined in these categories. It is also proven that a number of standardly used transformation operators in fuzzy sets, fuzzy rough sets, or fuzzy soft sets are in the form of general operators defined by monadic relations.

Monadic Power Set Theories in Kleisli Categories

“…In this paper, we continued to build relational variants of categories K, i.e., such modifications of a category K, where instead of K-morphisms f : X → Y of this category, special relations R : X Y defined by a monad T in this category are considered. Then Kleisli category K T of a category K defined by a monad T can be considered as a relational variant of a category K. We first presented this type of transformation of a category K to relational version in [11], where we showed that many standard fuzzy type relations in various categories are in fact relations defined by monads and, therefore, categories with such defined relations as morphisms are isomorphic to Kleisli category.…”

“…Example 2.3 [11] The monad G = (G, , ρ, 1 Set(L ) ) in the category Set(L ) is defined by 1. The object function G : Set(L ) → Set(L ) is defined by G(X, δ ) = (F(X, δ ), σ X,δ ), where the similarity relation σ is defined by…”

“…If f : X Y and g : Y Z are T-relations, their composition is a T-relation defined by g♦ f : X Z. This notion was introduced in the paper of Manes [10] and it was recently proved that it is an universal construction of relations for many fuzzy type structures (e.g., see [11]). Using the notion of a T-relation X Y , we can define a notion of a transformation of objects from T (X) to objects from T (Y ).…”

“…Example 2.4 [11] Recall that an L -valued fuzzy relation from X to Y is an L -valued fuzzy set R in a set X × Y . It is easy to see that R is an L -valued fuzzy relation if and only if R is a Z-relation X Y , where R(x)(y) = R(x, y) and Z is the monadic power set theory from Example 2.2.…”

“…This theory is the theory of monads in categories. Monads in a category were introduced by several authors, (for introduction see [4,9]) and have been used in fuzzifying mathematical objects, including power set structures of fuzzy sets [10,11,15,16] or fuzzy automata [1,2,3,12]. The idea of using monads for fuzzyfication is based on extension of objects X of a category K to another object T (X) ∈ K. These new objects may be regarded as objects of clouds of fuzzy states.…”