Geometrical isomorphisms between categories of fuzzy coverings and fuzzy partitions

TL;DR

This paper establishes geometrical isomorphisms between categories of fuzzy coverings and fuzzy partitions, enabling bijections between these structures with finitely many sets, thus deepening the categorical understanding of fuzzy set decompositions.

Contribution

It constructs categorical isomorphisms between fuzzy coverings and fuzzy partitions, revealing new bijections and structural insights in fuzzy set theory.

Findings

Established an isomorphism between categories of fuzzy coverings and partitions.

Derived bijections between fuzzy coverings and partitions with finitely many sets.

Provided a categorical framework for understanding fuzzy set decompositions.

Abstract

Let $Covering$ be the category of the category of fuzzy coverings, and $Partition$, the category of fuzzy partitions. We geometrically construct an isomorphism of categories between $Partition$ and a full subcategory of $Covering$, which can be used to derive bijections between fuzzy partitions and fuzzy coverings with finitely many sets. Also, we establish an isomorphism between $Covering[n]$, the category of coverings with $n$ fuzzy sets, and a subcategory of $Partition$, whose objects are partitions with $n$ sets which satisfy certain conditions, which can be also used to deduce another bijection between fuzzy partitions and fuzzy coverings with finitely many sets.