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On completely regular and Clifford ordered semigroups

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Abstract

Lee and Kwon (Sci Math 2:247–251, 1998) defined an ordered semigroup S to be completely regular if \(a \in (a^2Sa^2]\) for every \(a \in S\). We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green’s Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies \(e \le e^2\). Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed.

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References

  1. Cao, Y.: On weak commutativity of po-semigroups and their semilattice decompositions. Semigroup Forum 58, 386–394 (1999)

    Article  MathSciNet  Google Scholar 

  2. Cao, Y.: Chain decompositions of ordered semigroups. Semigroup Forum 65, 83–106 (2002)

    Article  MathSciNet  Google Scholar 

  3. Cao, Y., **nzhai, X.: Nil-extensions of simple po-semigroups. Commun. Algebra 28(5), 2477–2496 (2000)

    Article  MathSciNet  Google Scholar 

  4. Changphas, T.: Ascending chain conditions on principal left and right ideals for semidirect products of ordered semigroups. Quasigroup Relat. Syst. 21, 147–154 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Changphas, T., Phaipong, N.: On dual ordered semigroups. Quasigroup Relat. Syst. 22, 193–200 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Changphas, T., Luangchaisri, P., Mazurek, R.: On right chain ordered semigroups. Semigroup Forum 96, 523–535 (2018)

    Article  MathSciNet  Google Scholar 

  7. Gao, Z.: On the least property of the semilattice congruence on PO-semigroups. Semigroup Forum 56, 323–333 (1998)

    Article  MathSciNet  Google Scholar 

  8. Hansda, K., Jamader, A.: Characterization of inverse ordered semigroups by their ordered idempotents and bi-ideals. Quasigroup Relat. Syst. 28(1) (forthcoming)

  9. Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  10. Kehayopulu, N.: Remarks on ordered semigroups. Math. Jpn. 35, 1061–1063 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Kehayopulu, N.: Note on Green’s relation in ordered semigroup. Math. Jpn. 36, 211–214 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Kehayopulu, N.: On completely regular \(poe\)-semigroups. Math. Jpn. 37, 123–130 (1992)

    MATH  Google Scholar 

  13. Kehayopulu, N.: On regular duo ordered semigroups. Math. Jpn. 37, 535–540 (1992)

    MathSciNet  MATH  Google Scholar 

  14. Kehayopulu, N.: On intra-regular ordered semigroups. Semigroup Forum 46, 271–278 (1993)

    Article  MathSciNet  Google Scholar 

  15. Kehayopulu, N.: On completely regular ordered semigroups. Sci. Math. 1(1), 27–32 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Kehayopulu, N.: The role of the ideal elements in studying the structure of some ordered semigroups. Turk. J. Math. 41, 1144–1154 (2017)

    Article  MathSciNet  Google Scholar 

  17. Kehayopulu, N.: A note on archimedean ordered semigroups. Algebra Univers. 79, 27 (2018)

    Article  MathSciNet  Google Scholar 

  18. Lee, S.K., Kwon, Y.I.: On completely regular and quasi-completely regular ordered semigroups. Sci. Math. 2, 247–251 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Petrich, M., Reilly, N.: Completely Regular Semigroups. Wiley, New York (1999)

    MATH  Google Scholar 

  20. Saito, T.: Ordered idempotent semigroups. J. Math. Soc. Japan 14(2), 150–169 (1962)

    Article  MathSciNet  Google Scholar 

  21. Saito, T.: Regular elements in an ordered semigroup. Pac. J. Math. 13, 263–295 (1963)

    Article  MathSciNet  Google Scholar 

  22. Saito, T.: Ordered completely regular semigroups. Pac. J. Math. 14(1), 295–308 (1964)

    Article  MathSciNet  Google Scholar 

  23. Saito, T.: Ordered inverse semigroups. Trans. Am. Math. Soc. 153, 99–138 (1971)

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Visva-Bharati, Santiniketan, West Bengal, 731235, India

    A. K. Bhuniya & K. Hansda

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  1. A. K. Bhuniya
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  2. K. Hansda
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Correspondence to K. Hansda.

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Bhuniya, A.K., Hansda, K. On completely regular and Clifford ordered semigroups. Afr. Mat. 31, 1029–1045 (2020). https://doi.org/10.1007/s13370-020-00778-1

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