On inverse and right inverse ordered semigroups

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Abstract

A regular ordered semigroup \(S\) is called right inverse if every principal left ideal of \(S\) is generated by an \(\mathcal{R}\)-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element \(a\in S\) are \(\mathcal{R}\)-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.

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Ideals and Green's relations in ordered semigroups

Niovi Kehayopulu (2006)

Exactly as in semigroups, Green's relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the ℐ -trivial ordered semigroups which are defined via the Green's relation ℐ , and with the nil and Δ -ordered semigroups. We prove that every nil ordered semigroup is ℐ -trivial which means that there is no ordered semigroup which is 0-simple and nil at the same time. We show that in nil ordered semigroups which are chains with respect to the divisibility ordering, every complete congruence is a Rees congruence, and that this type of ordered semigroups are △ -ordered semigroups, that is, ordered semigroups for which the complete congruences form a chain. Moreover, the homomorphic images of △ -ordered semigroups are △ -ordered semigroups as well. Finally, we prove that the ideals of a nil ordered semigroup S form a chain under inclusion if and only if S is a chain with respect to the divisibility ordering.