### 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.

**How to Cite this Article:**Niovi Kehayopulu, On semisimple and left quasi-regular elements of ordered semigroups, Journal of Mathematical and Computational Science, Vol 3, No 1 (2013), 207-215 Copyright © 2013 Niovi Kehayopulu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

J. Math. Comput. Sci.

ISSN: 1927-5307

Editorial Office: jmcs@scik.org

Copyright ©2019 JMCS