Abstract
Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined.
Highlights
In fuzzy set theory many structures are used, which are based on various modifications of topological operators
The common feature of these categories was that the morphisms in these categories were based on mappings between the underlying sets of corresponding objects
A number of results have emerged in the theory of fuzzy sets, which are based on the application of fuzzy relations as morphisms in suitable categories
Summary
In fuzzy set theory many structures are used, which are based on various modifications of topological operators. These structures include variants of fuzzy topological spaces, fuzzy rough sets, fuzzy approximation spaces, fuzzy closure operators, fuzzy pretopological operators and their dual terms, such as fuzzy interior operators For examples of these structures see, e.g., [1,2,3,4,5,6,7,8]. The main results of this paper are Theorems 1 and 2, which prove the existence of 23 functors among these categories, including how these functors combine with each other It follows from these theorems that the key category between the above categories is the category of spaces with fuzzy partitions with special fuzzy relations as morphisms. The main result are two theorems identifying functors among these categories, which represent the transformation processes among the individual structures
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