On ordered Abel–Grassmann’s groupoids

The concept of $(m,n)$-ideals in ordered $\mathcal{AG}$-groupoids is introduced and the $(0,2)$-ideals and $(1,2)$-ideals of an ordered $\mathcal{AG}$-groupoid in terms of left ideals are characterised. It is shown that an ordered $\mathcal{AG}$-groupoid $S$ is $0$–$(0,2)$-bisimple if and only if $S$ is right $0$-simple. The results of this paper extend the concept of an $\mathcal{AG}$-groupoid without order. Finally, we characterize an intra-regular ordered $\mathcal{AG}$-groupoid in terms of left and right ideals.