# On left regular and intra-regular ordered semigroups

In: Mathematica Slovaca, vol. 64, no. 5
Niovi Kehayopulu - Michael Tsingelis

## Details:

Year, pages: 2014, 1123 - 1134
Keywords:
ordered semigroup, left regular, intra-regular, left ideal, semiprime, left simple, (complete) semilattice congruence, (complete) semilattice of left simple (left regular, simple) semigroups
About article:
We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: \textit{On left regular ordered semigroups}, Math. Japon. \textbf{35} (1990), 1057–1060] and [KEHAYOPULU, N.: \textit{On intra-regular ordered semigroups}, Semigroup Forum \textbf{46} (1993), 271–278]. We prove that an ordered semigroup $S$ is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of $S$. Moreover, $S$ is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of $S$. The right analog also holds. The same result is true if we replace the words left regular'' by intra-regular''. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.
How to cite:
ISO 690:
Kehayopulu, N., Tsingelis, M. 2014. On left regular and intra-regular ordered semigroups. In Mathematica Slovaca, vol. 64, no.5, pp. 1123-1134. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0263-1

APA:
Kehayopulu, N., Tsingelis, M. (2014). On left regular and intra-regular ordered semigroups. Mathematica Slovaca, 64(5), 1123-1134. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0263-1
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