On Fuzzy Ideals and Level Subsets of Ordered $\Gamma$-Groupoids

TL;DR

This paper characterizes fuzzy ideals and prime ideals in ordered mbda-groupoids using level subsets and explores their behavior under Cartesian products, establishing foundational properties for fuzzy ideal theory.

Contribution

It provides new characterizations of fuzzy ideals and prime ideals in ordered mbda-groupoids via level subsets and analyzes their Cartesian product properties.

Findings

Fuzzy ideals are characterized by their level subsets.

Cartesian product of fuzzy ideals remains a fuzzy ideal.

Level subsets of product fuzzy ideals retain ideal properties.

Abstract

We characterize the fuzzy left (resp. right) ideals, the fuzzy ideals and the fuzzy prime (resp. semiprime) ideals of an ordered $\Gamma$-groupoid $M$ in terms of level subsets and we prove that the cartesian product of two fuzzy left (resp. right) ideals of $M$ is a fuzzy left (resp. right) ideal of $M\times M$, and the cartesian product of two fuzzy prime (resp. semiprime) ideals of $M$ is a fuzzy prime (resp. semiprime) ideal of $M\times M$. As a result, if $\mu$ and $\sigma$ are fuzzy left (resp. right) ideals, ideals, fuzzy prime or fuzzy semiprime ideals of $M$, then the nonempty level subsets $(\mu\times\sigma)_t$ are so.