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Characterization of the Set of Regular Elements in Ordered Semigroups

Abstract We characterize the set of regular elements of an ordered semigroup S which is a left (right) ideal, an ideal, a bi-ideal or a quasi-ideal of S using the left (resp. right) ideal and the quasi-ideal LE(S)∩RE(S) of S, where E(S) is the set of elements of S for which e ≤ e2. Illustrative example is given.