A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups

Abstract

Mitsch defined the natural partial order \(\le \) on a semigroup S as follows: \(a\le b\) if and only if \(a=xb=by, a=xa\) for some \(x,y\in S^1\). Let \({{{\mathcal {T}}}}_X\) be the full transformation semigroup on a finite set \(X=\{1,2,\ldots ,n\}\). Let \(\rho \) be an equivalence relation on X and \(\preceq \) be a total order on the partition set \(X/\rho \) of X induced by \(\rho \). Denote by \({\overline{x}}\) the \(\rho \)-class containing \(x\in X\). In this paper, we endow the partition order-decreasing transformation subsemigroup of \({{{\mathcal {T}}}}_X\) defined by

$$\begin{aligned} T(\rho ,\preceq )=\{f\in {{{\mathcal {T}}}}_X: \forall \,\,x\in X, \overline{f(x)}\preceq {\overline{x}}\} \end{aligned}$$

with the natural partial order and give a characterization for this order. Then we determine the compatibility of their elements and find all the minimal and maximal elements.

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References

  1. 1.

    Pin, J.E.: Varieties of Formal Languages. North Oxford Academic Publishers Ltd, London (1986)

    Book  Google Scholar 

  2. 2.

    Chaopraknoi, S., Phongpattanacharoen, T., Prakitsri, P.: The natural partial order on linear semigroups with nullity and co-rank bounded below. Bull. Aust. Math. Soc. 91(1), 104–115 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chaopraknoi, S., Phongpattanacharoen, T., Rawiwan, P.: The natural partial order on some transformation semigroups. Bull. Aust. Math. Soc. 89(2), 272–292 (2014)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Hartwig, R.: How to partially order regular elements. Math. Jpn. 25(1), 1–13 (1980)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Kowol, G., Mitsch, H.: Naturally ordered transformation semigroups. Monatsh. Math. 102(2), 115–138 (1986)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Marques-Smith, M.P.O., Sullivan, R.P.: Partial orders on transformation semigroups. Monatsh. Math. 140(2), 103–118 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Mitsch, H.: A natural partial order for semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Nambooripad, K.S.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23(2), 249–260 (1980)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Pei, H.S., Deng, W.N.: Naturally ordered semigroups of partial transformations preserving an equivalence relation. Commun. Algebra 41(9), 3308–3324 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Singha, B., Sanwong, J., Sullivan, R.P.: Partial order on partial Baer–Levi semigroups. Bull. Aust. Math. Soc. 81(2), 197–206 (2010)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Sullivan, R.P.: Partial orders on linear transformation semigroups. Proc. R. Soc. Edinb. A 135(2), 413–437 (2005)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Sun, L., Pei, H.S., Cheng, Z.X.: Naturally ordered transformation semigroups preserving an equivalence. Bull. Aust. Math. Soc. 78(2), 117–128 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Sun, L., Deng, W.N., Pei, H.S.: Naturally ordered transformation semigroups preserving an equivalence and a cross-section. Algebr. Colloq. 18(3), 523–532 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Sun, L., Wang, L.M.: Natural partial order in semigroups of transformations with invariant set. Bull. Aust. Math. Soc. 87(1), 94–107 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Sun, L., Sun, J.L.: A natural partial order on certain semigroups of transformations with restricted range. Semigroup Forum 92(1), 135–141 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Han, X.F., Sun, L.: A natural partial order on certain semigroups of transformations restricted by an equivalence. Bull. Iran. Math. Soc. 44(6), 1571–1579 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Umar, A.: On the semigroups of order-decreasing finite full transformations. Proc. R. Soc. Edinb. A 120(1–2), 129–142 (1992)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Wagner, V.: Generalized groups. Dokl. Akad. Nauk SSSR 84, 1119–1122 (1952). (in Russian)

    MathSciNet  Google Scholar 

  19. 19.

    Yang, X.L., Yang, H.B.: Maximal half-transitive submonoids of full transformation semigroups. Adv. Math. (in Chinese) 40(5), 580–586 (2011)

    MathSciNet  Google Scholar 

  20. 20.

    Yang, H.B., Yang, X.L.: Automorphisms of partition order-decreasing transformation monoids. Semigroup Forum 85(3), 513–524 (2012)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgement

I would like to thank the referee for his/her valuable suggestions and comments which helped to improve the presentation of this paper.

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Correspondence to Lei Sun.

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Sun, L. A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups. Bull. Iran. Math. Soc. 46, 1357–1369 (2020). https://doi.org/10.1007/s41980-019-00329-w

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Keywords

  • Transformation semigroup
  • Natural partial order
  • Compatibility
  • The minimal (maximal) elements

Mathematics Subject Classification

  • 20M20