Conditions that the quotient of $L$-fuzzy up-sets forms a complete lattice

TL;DR

This paper establishes conditions under which the quotient of $L$-fuzzy up-sets forms a complete lattice, using closure operators, and shows isomorphism to certain intervals.

Contribution

It provides a necessary and sufficient condition for the quotient of $L$-fuzzy up-sets to form a complete lattice and relates it to interval isomorphisms.

Findings

A condition for representing families of subsets by $L$-fuzzy up-sets.

Necessary and sufficient condition for the quotient to be a complete lattice.

The quotient of certain $L$-fuzzy up-sets is isomorphic to an interval.

Abstract

This paper deals with conditions under which the quotient of $L$-fuzzy up-sets forms a complete lattice by using terminologies of closure operators. It first gives a condition that a family of some subsets of a nonempty set can be represented by $L$-fuzzy up-sets, which is then used to formulate a necessary and sufficient condition under which the quotient of $L$-fuzzy up-sets forms a complete lattice. This paper finally shows that the quotient of a kind of $L$-fuzzy up-sets is isomorphic to an interval generated by an $L$-fuzzy up-set.