Regularity of Semigroups in Terms of Hybrid Ideals and Hybrid Bi-Ideals

Balasubramanian Elavarasan, Young Bae Jun

In this paper, we establish some equivalent conditions for a semigroup to be regular and intra-regular, in terms of hybrid ideals and hybrid bi-ideals. We also characterize the left and right simple and the completely regular semigroups utilizing hybrid ideals and hybrid bi-ideals. We show that a semigroup $\mathcal{S}$ is left simple if and only if it is hybrid left simple. We also prove that if a semigroup $\mathcal{S}$ is intra-regular, then for each hybrid ideal $\tilde{j}_µ$ of $\mathcal{S},$ we have $\tilde{j}_µ(r_1r_2)=\tilde{j}_µ(r_2r_1)$ for all $r_1,r_2\in \mathcal{S}$.