Properties of ordered AG-groupoids by their hybrid ideals

Abstract

In this paper, we define hybrid ideals and hybrid bi-ideals of ordered $\mathcal{AG}-$groupoids $\mathbb{Y}$ and investigate their properties in terms of regular, intra-regular and weakly regular of $\mathbb{Y}$. We build appropriate examples to demonstrate that not every hybrid generalised bi-ideal of $\mathbb{Y}$ is a hybrid bi-ideal of $\mathbb{Y}$ and that not every hybrid (1,2)-ideal of $\mathbb{Y}$ is a hybrid left (resp., right) ideal of $\mathbb{Y}$. We also obtain equivalent conditions for an ordered $\mathcal{AG-}$groupoid $\mathbb{Y}$ to be intra-regular and left weakly regular.