On the decomposition of completely regular ordered semigroups into union of left and right simple ordered semigroups

Michael Tsingelis

Abstract


We give sufficient conditions (Rs-condition and Ls-condition) which a completely regular ordered semigroup S must satisfy so that the set Ge={a∈S|a∈(eS]∩(Se],e∈(aS]∩(Sa]},e∈E(S), is (maximal under the inclusion relation) right and left simple subsemigroup of S, where E(S) is the set of elements of S for which e≤e2. We prove that S=Ue∈E(S)Ge and thus S is decomposed into a union of (disjoint) right and left simple semigroups. In addition every N-class of a completely regular ordered semigroup satisfying Rs-condition and Ls-condition is a union of right and left simple subsemigroups of S, where N is the least complete semilattice congruence on S. Finally we prove that the previous decompositions into unions of right and left simple semigroups characterize equivalently an ordered semigroup satisfying both Rs-condition and Ls-condition in order to be a completely regular ordered semigroup.


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Published: 2022-02-18

How to Cite this Article:

Michael Tsingelis, On the decomposition of completely regular ordered semigroups into union of left and right simple ordered semigroups, J. Semigroup Theory Appl., 2022 (2022), Article ID 1

Copyright © 2022 Michael Tsingelis. 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.

Journal of Semigroup Theory and Applications

ISSN 2051-2937

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