Novel fuzzy topologies from old through fuzzy primals

TL;DR

This paper introduces the concept of fuzzy primals and primal fuzzy topologies, exploring their properties, operators, and relationships to extend fuzzy topology theory.

Contribution

It proposes a new fuzzy structure called fuzzy primal, along with operators and methods to generate and analyze primal fuzzy topologies.

Findings

Defined fuzzy primal and primal fuzzy topology.

Introduced fuzzy operators (.)$^ riangle$ and Cl$^ riangle$ for topology construction.

Established conditions for compatibility between fuzzy primals and topologies.

Abstract

In this paper, we introduce a novel fuzzy structure named "fuzzy primal". We study the essential properties and discuss basic operations on it. A fuzzy operator (.)$^\diamond$ on the family of all fuzzy sets is introduced here by applying the q-neighborhood structure to a primal fuzzy topological space along with the Lukasiewicz disjunction. We explore the main characterizations of (.)$^\diamond$. Then, we define another fuzzy operator, symbolized by Cl$^\diamond$, with the utilization of (.)$^\diamond$. These fuzzy operators are studied in order to deduce a new fuzzy topology from the original one. Such a new fuzzy topology is called primal fuzzy topology. The fundamental structure, particularly a fuzzy base that generates primal fuzzy topologies, as well as many relationships between different fuzzy primals and fuzzy topologies, are also analyzed. Lastly, the concept of compatibility between fuzzy primals and fuzzy topologies is introduced, and some equivalent conditions related to this are examined. It is shown that if a fuzzy primal is compatible with a fuzzy topology, then the fuzzy base that generates the primal fuzzy topology is itself a fuzzy topology.