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

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 S is left simple if and only if it is hybrid left simple. We also prove that if a semigroup S is intra-regular, then for each hybrid ideal ˜jµ of S, we have ˜jµ(r1r2) = ˜jµ(r2r1) for all r1, r2 ∈ S.