On semisimple and left quasi-regular elements of ordered semigroups

Abstract

We prove, among others, that an ordered semigroup contains a left (resp. right) quasi-regular element if and only it contains a left (resp. right) regular element. It has a semisimple element if and only if it has an intra-regular element. An element a of an ordered semigroup S is a semisimple element of S if and only if there exists an intra-regular element b of S such that I(a) = I(b). The element a is a left (resp. right) quasi-regular element of S if and only if there exists a left (resp. right) regular element b of S such that L(a) = L(b) (resp. R(a) = R(b)). As a consequence, if the ideal I(a) generated by an element a of S has an intra-regular generator, then a is semisimple. If the principal left (resp. right) ideal L(a) (resp. R(a)) of an element a of S has a left (resp. right) regular generator, then a is a left (resp. right) quasi-regular element of S.