Characterizations of right modular groupoids by $(\in, \in \vee q_{k})$-fuzzy ideals

TL;DR

This paper introduces and characterizes various types of $( ext{in}, ext{in} ext{vee} q_k)$-fuzzy ideals in right modular groupoids, establishing their equivalences in completely regular cases.

Contribution

It defines $( ext{in}, ext{in} ext{vee} q_k)$-fuzzy ideals and proves their equivalence in completely regular right modular groupoids, advancing the algebraic theory of fuzzy ideals.

Findings

$( ext{in}, ext{in} ext{vee} q_k)$-fuzzy ideals are introduced.

Equivalence of different fuzzy ideals in completely regular right modular groupoids is established.

Provides new characterizations of regularity using fuzzy ideals.

Abstract

In this paper, we have introduced the concept of $\left( \in ,\in \vee q\right) $-fuzzy ideals in a right modular groupoid. We have discussed several important features of a completely regular right modular groupoid by using the $\left( \in ,\in \vee q\right) $-fuzzy left (right, two-sided) ideals, $\left( \in ,\in \vee q\right) $-fuzzy (generalized) bi-ideals and $\left( \in ,\in \vee q\right) $-fuzzy $(1,2)$-ideals. We have also used the concept of $\left( \in ,\in \vee q_{k}\right) $-fuzzy left (right, two-sided) ideals, $\left( \in ,\in \vee q_{k}\right) $-fuzzy quasi-ideals $\left( \in ,\in \vee q_{k}\right) $-fuzzy bi-ideals and $% \left( \in ,\in \vee q_{k}\right) $-fuzzy interior ideals in completely regular right modular groupoid and proved that the $\left( \in ,\in \vee q_{k}\right) $-fuzzy left (right, two-sided), $\left( \in ,\in \vee q_{k}\right) $-fuzzy (generalized) bi-ideals, and $\left( \in ,\in \vee q_{k}\right) $-fuzzy interior ideals coincide in a completely regular right modular groupoid.