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On Hartwig–Nambooripad orders

  • Alexander GutermanEmail author
  • Xavier Mary
  • Pavel Shteyner
Research Article
  • 61 Downloads

Abstract

We obtain several equivalent characterizations for the Hartwig–Nambooripad order based on outer inverses. We also introduce new partial orders on arbitrary semigroups extending these regular semigroup orders. On epigroups these new partial orders admit characterizations in terms of outer inverses.

Keywords

Generalized inverses Hartwig–Nambooripad order Ordered semigroups 

Notes

Acknowledgements

The authors wish to thank the referee for careful reading.

References

  1. 1.
    Cen, J.M.: On the unified theory of matrix partial orders. Gongcheng Shuxue Xuebao 22(1), 139–143 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Mathematical Surveys, vol. I. AMS, Providence (1961)Google Scholar
  3. 3.
    Drazin, M.P.: Pseudo-inverse in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Drazin, M.P.: A partial order in completely regular semigroups. J. Algebra 98(2), 362–374 (1986)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Green, J.A.: On the structure of semigroups. Ann. Math. (Second Ser.) 54(1), 163–172 (1951)CrossRefGoogle Scholar
  6. 6.
    Guterman, A., Johnson, M., Kambites, M.: Linear isomorphisms preserving Green’s relations for matrices over anti-negative semifields. Linear Algebra Appl. 545, 1–14 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hartwig, R.E.: How to partially order regular elements. Math. Japonica 25, 1–13 (1980)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Higgins, P.M.: The Mitsch order on a semigroup. Semigroup Forum 49(2), 261–266 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs. New Series, vol. 12. Oxford Science Publications, Oxford (1995)Google Scholar
  10. 10.
    Kaplansky, I.: Rings of operators. W. A. Benjamin Inc, New York, Amsterdam (1968)zbMATHGoogle Scholar
  11. 11.
    Lyapin, E.S.: Semigroups. Izdat. Fiz.-Mat. Lit., Moscow (1960). (in Russian)zbMATHGoogle Scholar
  12. 12.
    Mitsch, H.: A natural partial order on semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mitsch, H.: Semigroups and their natural order. Math. Slovaca 44(4), 445–462 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mitsch, H.: Classes of semigroups with compatible natural partial order I. Math. Pannon. 22(2), 165–198 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mitsch, H.: Classes of semigroups with compatible natural partial order II. Math. Pannon. 23(1), 9–43 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mitra, S.K.: The minus partial order and the shorted matrix. Linear Algebra Appl. 83, 1–27 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mitra, S.K., Hartwig, R.E.: Partial orders based on outer inverse. Linear Algebra Appl. 176, 3–20 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Munn, W.D.: Pseudo-inverses in semigroups. Math. Proc. Camb. Philos. Soc. 57(2), 247–250 (1961)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nambooripad, K.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Petrich, M.: Certain partial orders on semigroups. Czechoslov. Math. J. 51, 415–432 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rakic, D.S., Djordjevic, D.S.: Partial orders in rings based on generalized inverses—unified theory. Linear Algebra Appl. 471, 203–223 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vagner, V.V.: Generalized groups. Doklady Akad. Nauk SSSR (N.S.) 84, 1119–1122 (1952). (in Russian)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Alexander Guterman
    • 1
    • 2
    • 3
    Email author
  • Xavier Mary
    • 4
  • Pavel Shteyner
    • 1
    • 2
    • 3
  1. 1.Department of Mathematics and MechanicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Moscow Center for Continuous Mathematical EducationMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  4. 4.Modal’X, UPLUniversity Paris NanterreNanterreFrance

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