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

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.