Abstract
In this paper, we characterize right weakly regular le-semigroups in terms of its right-ideal elements, ideal elements, bi-ideal elements, generalized bi-ideal elements and interior-ideal elements. We also provide some sufficient conditions on poe-semigroups under which these elements coincide with each other. We, then, introduce the notions of left (resp. right) weakly prime, left (resp. right) weakly semiprime, weakly prime bi-ideal and weakly semiprime bi-ideal elements in poe-semigroups and prove some results concerning weakly prime bi-ideal and weakly semiprime bi-ideal elements. We also extend the concept of an m-system and an n-system in a poe-semigroup and relate these systems with weakly prime and weakly semiprime ideal-elements of ve-semigroups. Finally, we show that, in any le-semigroup, either a B-class (resp. Q-class, H-class) is right weakly regular or none of its element is right weakly regular.