On the uniqueness of $L$-fuzzy sets in the representation of families of sets

TL;DR

This paper investigates the conditions under which $L$-fuzzy sets uniquely represent families of subsets, providing formulas for their count and criteria for uniqueness.

Contribution

It introduces a formula for counting $L$-fuzzy sets with a given collection of cuts and establishes a necessary and sufficient condition for their uniqueness.

Findings

Derived a formula for the number of $L$-fuzzy sets with a specified collection of cuts.

Established a necessary and sufficient condition for the uniqueness of $L$-fuzzy sets in representation.

Provided theoretical insights into the structure of $L$-fuzzy sets and their representations.

Abstract

This paper deals with the uniqueness of $L$-fuzzy sets in the representation of a given family of subsets of nonempty set. It first shows a formula of the number of $L$-fuzzy sets whose collection of cuts coincides with a given family of subsets of a nonempty set, and then provides a necessary and sufficient condition under which such $L$-fuzzy sets are unique.