Research Article
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Year 2022, Volume: 71 Issue: 3, 769 - 777, 30.09.2022
https://doi.org/10.31801/cfsuasmas.1019458

Abstract

References

  • Adenji, A.O., Science, C., Makanjuola, S.O., On some combinatorial results of collapse and properties of height in full transformation semigroups, Afr. J. Comp. ICT., 1(2) (2008), 61-63.
  • Adenji, A.O., Science, C., Makanjuola, S.O., A combinatorial property of the full transformation semigroup, Afr. J. Comp. ICT., 2(1) (2009), 15-19.
  • Ayık, G., Ayık, H., Koc, M., Combinatorial results for order-preserving and order-decreasing transformations, Turk. J. Math., 35(4) (2011), 1-9. https://doi.org/10.3906/mat-1010-432
  • Catarino P. M., Higgins, P.M., The monoid of orientation-preserving mappings on a chain, Semigroup Forum, 58(2) (1999), 190-206. https://doi.org/10.1007/s002339900014
  • Clifford, A.H., Preston, G.B., The algebraic theory of semigroups. Vol.II., American Math. Soc., Providence, 1967.
  • Ganyushkin, O., Mazorchuk, V., Classical Finite Transformation Semigroups, Springer-Verlag, London, 2009.
  • Grimaldi R.P., Discrete and combinatorial mathematics, Pearson Education Inc., USA, 2003.
  • Higgins, P.M., Combinatorial results for semigroups of order-preserving mappings, Math. Proc. Cambridge Phil. Soc., 113(2) (1993), 281-296. https://doi.org/10.1017/S0305004100075964
  • Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc., 41(1) (1966), 707-716. https://doi.org/10.1112/jlms/s1-41.1.707
  • Howie, J. M., Products of idempotents in certain semigroups of transformations, Proc. Edinb. Math. Soc., 17(3) (1970/71), 223-236. https://doi.org/10.1017/S0013091500026936
  • Howie, J.M., Fundamentals of Semigroup Theory, Oxford University Press, New York, 1995.
  • Korkmaz, E., Ayık, H., Ranks of nilpotent subsemigroups of order-preserving and decreasing transformation semigroups, Turk. J. Math., 45(4) (2021), 1626-1634. https://doi.org/10.3906/mat-2008-19
  • Laradji, A., Umar, A., On certain finite semigroups of order-decreasing transformations I, Semigroup Forum, 69(2) (2004), 184-200. https://doi.org/10.1007/s00233-004-0101-9
  • Mbah, M. A., Ndubisi, R. U., Achaku, D. T., On some combinatorial results of collapse in partial transformation semigroups, Canadian Journal of Pure and Applied Sciences, 1, (2020) 257-261. https://doi.org/10.15864/jmscm.1301
  • Umar, A., Semigroups of order-decreasing transformations: The isomorphism theorem, Semigroup Forum, 53(1) (1996), 220–224. https://doi.org/10.1007/BF02574137
  • Yağcı, M., Korkmaz, E., On nilpotent subsemigroups of the order-preserving and decreasing transformation semigroups, Semigroup Forum, 101(2) (2020), 486-289. https://doi.org/10.1007/s00233-020-10098-2

Combinatorial results of collapse for order-preserving and order-decreasing transformations

Year 2022, Volume: 71 Issue: 3, 769 - 777, 30.09.2022
https://doi.org/10.31801/cfsuasmas.1019458

Abstract

The full transformation semigroup TnTn is defined to consist of all functions from Xn={1,,n}Xn={1,…,n} to itself, under the operation of composition. In \cite{JMH1}, for any αα in TnTn, Howie defined and denoted collapse by c(α)=t\im(α){tα1:|tα1|2}c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}. Let OnOn be the semigroup of all order-preserving transformations and CnCn be the semigroup of all order-preserving and decreasing transformations on Xn
Xn=
under its natural order, respectively.
Let E(On)E(On) be the set of all idempotent elements of OnOn, E(Cn)E(Cn) and N(Cn)N(Cn) be the sets of all idempotent and nilpotent elements of CnCn, respectively. Let UU be one of {Cn,N(Cn),E(Cn),On,E(On)}{Cn,N(Cn),E(Cn),On,E(On)}. For αUα∈U, we consider the set
\imc(α)={t\im(α):|tα1|2}\imc(α)={t∈\im(α):|tα−1|≥2}. For positive integers 2krn2≤k≤r≤n, we define
U(k)={αU:t\imc(α) and |tα1|=k},U(k,r)={αU(k):t\imc(α)tα1|=r}.U(k)={α∈U:t∈\imc(α) and |tα−1|=k},U(k,r)={α∈U(k):|⋃t∈\imc(α)tα−1|=r}.
The main objective of this paper is to determine |U(k,r)||U(k,r)|, and so |U(k)||U(k)| for some values rr and kk.

References

  • Adenji, A.O., Science, C., Makanjuola, S.O., On some combinatorial results of collapse and properties of height in full transformation semigroups, Afr. J. Comp. ICT., 1(2) (2008), 61-63.
  • Adenji, A.O., Science, C., Makanjuola, S.O., A combinatorial property of the full transformation semigroup, Afr. J. Comp. ICT., 2(1) (2009), 15-19.
  • Ayık, G., Ayık, H., Koc, M., Combinatorial results for order-preserving and order-decreasing transformations, Turk. J. Math., 35(4) (2011), 1-9. https://doi.org/10.3906/mat-1010-432
  • Catarino P. M., Higgins, P.M., The monoid of orientation-preserving mappings on a chain, Semigroup Forum, 58(2) (1999), 190-206. https://doi.org/10.1007/s002339900014
  • Clifford, A.H., Preston, G.B., The algebraic theory of semigroups. Vol.II., American Math. Soc., Providence, 1967.
  • Ganyushkin, O., Mazorchuk, V., Classical Finite Transformation Semigroups, Springer-Verlag, London, 2009.
  • Grimaldi R.P., Discrete and combinatorial mathematics, Pearson Education Inc., USA, 2003.
  • Higgins, P.M., Combinatorial results for semigroups of order-preserving mappings, Math. Proc. Cambridge Phil. Soc., 113(2) (1993), 281-296. https://doi.org/10.1017/S0305004100075964
  • Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc., 41(1) (1966), 707-716. https://doi.org/10.1112/jlms/s1-41.1.707
  • Howie, J. M., Products of idempotents in certain semigroups of transformations, Proc. Edinb. Math. Soc., 17(3) (1970/71), 223-236. https://doi.org/10.1017/S0013091500026936
  • Howie, J.M., Fundamentals of Semigroup Theory, Oxford University Press, New York, 1995.
  • Korkmaz, E., Ayık, H., Ranks of nilpotent subsemigroups of order-preserving and decreasing transformation semigroups, Turk. J. Math., 45(4) (2021), 1626-1634. https://doi.org/10.3906/mat-2008-19
  • Laradji, A., Umar, A., On certain finite semigroups of order-decreasing transformations I, Semigroup Forum, 69(2) (2004), 184-200. https://doi.org/10.1007/s00233-004-0101-9
  • Mbah, M. A., Ndubisi, R. U., Achaku, D. T., On some combinatorial results of collapse in partial transformation semigroups, Canadian Journal of Pure and Applied Sciences, 1, (2020) 257-261. https://doi.org/10.15864/jmscm.1301
  • Umar, A., Semigroups of order-decreasing transformations: The isomorphism theorem, Semigroup Forum, 53(1) (1996), 220–224. https://doi.org/10.1007/BF02574137
  • Yağcı, M., Korkmaz, E., On nilpotent subsemigroups of the order-preserving and decreasing transformation semigroups, Semigroup Forum, 101(2) (2020), 486-289. https://doi.org/10.1007/s00233-020-10098-2
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Emrah Korkmaz 0000-0002-4085-0419

Publication Date September 30, 2022
Submission Date November 5, 2021
Acceptance Date March 29, 2022
Published in Issue Year 2022 Volume: 71 Issue: 3

Cite

APA Korkmaz, E. (2022). Combinatorial results of collapse for order-preserving and order-decreasing transformations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(3), 769-777. https://doi.org/10.31801/cfsuasmas.1019458
AMA Korkmaz E. Combinatorial results of collapse for order-preserving and order-decreasing transformations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2022;71(3):769-777. doi:10.31801/cfsuasmas.1019458
Chicago Korkmaz, Emrah. “Combinatorial Results of Collapse for Order-Preserving and Order-Decreasing Transformations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 3 (September 2022): 769-77. https://doi.org/10.31801/cfsuasmas.1019458.
EndNote Korkmaz E (September 1, 2022) Combinatorial results of collapse for order-preserving and order-decreasing transformations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 3 769–777.
IEEE E. Korkmaz, “Combinatorial results of collapse for order-preserving and order-decreasing transformations”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 3, pp. 769–777, 2022, doi: 10.31801/cfsuasmas.1019458.
ISNAD Korkmaz, Emrah. “Combinatorial Results of Collapse for Order-Preserving and Order-Decreasing Transformations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/3 (September 2022), 769-777. https://doi.org/10.31801/cfsuasmas.1019458.
JAMA Korkmaz E. Combinatorial results of collapse for order-preserving and order-decreasing transformations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:769–777.
MLA Korkmaz, Emrah. “Combinatorial Results of Collapse for Order-Preserving and Order-Decreasing Transformations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 3, 2022, pp. 769-77, doi:10.31801/cfsuasmas.1019458.
Vancouver Korkmaz E. Combinatorial results of collapse for order-preserving and order-decreasing transformations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(3):769-77.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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