# The Notions of Fuzzy Set on Pseudo Quasi Ordered Residuated Systems

## TL;DR

This paper explores fuzzy filters in quasi-ordered and pseudo quasi-ordered residuated systems and their properties.

## Contribution

It introduces and analyzes various types of fuzzy filters and their relationships in these systems.

## Key findings

- Comparative fuzzy filters in K can be normal under certain conditions.
- The set of all comparative fuzzy filters in K forms a complete lattice.
- Associative fuzzy filters in K are also implicative and comparative.

## Abstract

This paper introduces the concept of fuzzy filters and 2-fuzzy filters of a quasi-ordered residuated system

K
. This research explores comparative, normal, implicative and associative fuzzy filters within a quasi-ordered residuated system. It establishes the sufficient conditions under which a comparative fuzzy filter in

K
qualifies as a normal fuzzy filter and vice-versa. Furthermore, the study demonstrates that the collection of all comparative fuzzy filters in

K
constitutes a complete lattice. The notion of fuzzy filters of a pseudo quasi-ordered residuated system is introduced and the analysis extends to fantastic, comparative, associative and implicative fuzzy filters within a pseudo quasi-ordered residuated framework. Finally, it is proven that an associative fuzzy filter in

K
inherently satisfies the criteria for both implicative and comparative fuzzy filters.