Ideals and Green's relations in ordered semigroups

Niovi Kehayopulu

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-8

Abstract:

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 ℠-trivial ordered semigroups which are defined via the Green's relation ℠, and with the nil and Δ -ordered semigroups. We prove that every nil ordered semigroup is ℠-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.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:061286

DOI: 10.1155/IJMMS/2006/61286

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