Natural partial order and finiteness conditions on semigroups of linear transformations with invariant subspaces

  • Yanisa ChaiyaEmail author
Research Article


Given a vector space V and a subspace W of V, we consider the semigroup (under composition) S(VW), consisting of all linear transformations on V which leave the subspace W invariant. In this paper, we characterize the natural partial order on S(VW). In the partially ordered set, we determine the compatibility of their elements, and find all maximal and minimal elements. Also, we give necessary and sufficient conditions for S(VW) to be factorizable, unit-regular and directly finite.


Linear transformation semigroup Invariant subspace Natural partial order Finiteness condition 



  1. 1.
    Alarcao, H.D.: Factorizable as a finiteness condition. Semigroup Forum 20, 281–282 (1980)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boonmee, A.: Factorizable on some semigroup. M.S. thesis, Chiang Mai University, Chiang Mai (2007)Google Scholar
  3. 3.
    Chaiya, Y., Pookpienlert, C., Sanwong, J.: Semigroups of linear transformations with fixed subspaces: green’s relations, ideals and finiteness conditions. Asian Eur. J. Math.
  4. 4.
    Chaiya, Y., Honyam, P., Sanwong, J.: Maximal subsemigroups and finiteness conditions on transformation semigroups with fixed sets. Turk. J. Math. 41, 43–54 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, S.Y., Hsieh, S.C.: Factorizable inverse semigroups. Semigroup Forum 8, 283–297 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choomanee, W., Honyam, P., Sanwong, J.: Regularity in semigroup of transformations with invariant sets. Int. J. Pure Appl. Math. 87, 289–300 (2013)CrossRefGoogle Scholar
  7. 7.
    Ehrlich, G.: Unit-regular rings. Portugal. Math. 27, 209–212 (1968)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hartwig, R.: How to partially order regular elements. Math. Japn. 35, 1–13 (1980)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Honyam, P., Sanwong, J.: Semigroups of linear transformations with invariant subspace. Int. J. Algebra 6(8), 375–386 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Honyam, P., Sanwong, J.: Semigroups of transformations with invariant set. J. Korean Math. Soc. 48(2), 289–300 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematics Society Monographs, New Series, vol. 12. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  12. 12.
    Huisheng, P.: A note on semigroups of linear transformations with invariant subspace. Int. J. Algebra 6(27), 1319–1324 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jampachon, P., Saichalee, M., Sullivan, R.P.: Locally factorisable transformation semigroups. Southeast Asian Bull. Math. 25, 233–244 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kemprasit, Y.: Regularity and unit-regularity of generalized semigroups of linear transformations. Southeast Asian Bull. Math. 25, 617–622 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kowol, G., Mitsch, H.: Naturally ordered transformation semigroups. Monatsh. Math. 102, 115–138 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Magill Jr., K.D.: Subsemigroups of \(S(X)\). Math. Jpn. 11, 109–115 (1966)zbMATHGoogle Scholar
  17. 17.
    Marques-Smith, M.P.O., Sullivan, R.P.: Partial orders on transformation semigroups. Monatsh. Math. 140(2), 103–118 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mitsch, H.: A natural partial order for semigroup. Proc. Am. Soc. Edinb. 97(3), 384–388 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nambooripad, K.S.: The natural partial order on a regular semigroup. Proc. R. Soc. Edinb. 23, 249–260 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nenthein, S., Kemprasit, Y.: On transformation semigroups which are \({\cal{BQ}}\)-semigroup. Pure Math. Appl. 16(3), 307–314 (2006)zbMATHGoogle Scholar
  21. 21.
    Nenthein, S., Youngkhong, P., Kemprasit, Y.: Regular elements of some transformation semigroups. Pure Math. Appl. 16(3), 307–314 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Roman, S.: Advanced Linear Algebra. Graduate Texts in Mathematics, 3rd edn. Springer, Berlin (2008)CrossRefGoogle Scholar
  23. 23.
    Sullivan, R.P.: Partial orders on linear transformation semigroups. Proc. R. Soc. Edinb. 135A, 413–437 (2005)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sun, L., Wang, L.: Natural partial order in semigroups of transformations with invariant set. Bull. Aust. Math. Soc. 87(1), 94–107 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tirasupa, Y.: Factorizable transformation semigroup. Semigroup Forum 18, 15–19 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of Science and TechnologyThammasat University (Rangsit Campus)Pathum ThaniThailand

Personalised recommendations