On Rank of Semigroup of Order-Preserving Order-Decreasing Partial Contraction Mappings on a Finite Chain
Keywords:
Transformation semigroup, Contraction mappings, Order-preserving, Generating sets, Ideals
Abstract
Let [n] = {1, 2, . . . , n} be a finite chain, and ODCPn be the semigroup of order-preserving and order-decreasing partial contraction mappings on [n]. In this paper, we study the rank properties of the two-sided ideals of ODCPn. We show that the rank of Kp = {α ∈ ODCPn : |im α| ≤ p}, for 2 ≤ p ≤ n, is and hence, the rank of ODCPn is 2n
References
[1] Ali, B., Jada, M. A., Zubairu, M. M. (2018). On The Ranks Of Certain Semigroups Of OrderPreserving Partial Isometries of a Finite Chain. Journal of Algebra and Related Topics, 6(2): 15–33.
[2] Ali, B., Jada, M. A., Zubairu, M. M. (2023). Nilpotents in semigroups of order-preserving partial contraction mappings of a finite chain, Submitted.
[3] Al-Kharousi, F., Kehinde, R., Umar, A. (2014). Combinatorial results for certain semigroups of partial isometries of a finite chain. Australasian Journal of Combinatorics, 58(3): 365–375.
[4] Al-Kharousi, F., Kehinde, R., Umar, A. (2016). On the semigroup of partial isometries of a finite chain. Communications in Algebra, 44(2): 639–647.
[5] Ayik, G. Ayik, H. Howie, J. M., Ünlü, Y. (2008). Rank properties of semigroup of singular transformation on a finite set. Communications in Algebra, 36(7):2581–2587.
[6] Bugay, L. (2020). On the ranks of certain ideals of monotone contractions. Hacettepe Journal of Mathematics and Statistics, 49(6): 1988–1996
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[16] Green, J. A. (1951). On the structure of semigroups. Annals of Mathematics, 54(2):163–172.
[17] Howie, J. M., McFadden, R. B. (1990). Idempotent rank in finite full transformation semigroups, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, 114: 161-167.
[18] Howie, J. M. (1995). Fundamentals of semigroup theory. London Mathematical Society, New series 12. The Clarendon Press, Oxford University Press.
[19] Higgins, P. M. (1992). Techniques of semigroup theory. Oxford university Press.
[20] Imam, A. T., Usman, L., Idris, l., Ibrahim, S. (2024). Quasi-idempotent in finite semigroup of partial order-preserving transformation. International Journal of Mathematical Sciences and Optimization: Theory and Application, 10(1):53–59.
[21] Levi, I. (2006). Nilpotent Ranks of semigroups of partial transformations. Semigroup Forum, 72: 459–476
[22] Ladraji, A., Umar A. (2004). Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra, 278: 342–359.
[23] Peter, I. R., Mogbonju, M. M., Adeniji, A. O., Akinwunmi, S. A., Ibrahim, A. (2024). Some geometric characterization of star-like 3D conjugacy C3ωn∗ on partial one-one transformation semigroups. International Journal of Mathematical Sciences and Optimization: Theory and Application, 10(3):57–66.
[24] Toker K. (2020). Ranks of some subsemigroups of full contraction mappings on a finite chain. Journal of Balikesir University Institute of Science and Technology, 22(2): 403–414.
[25] Umar, A., Al-Kharousi, F. (2012). Studies in semigroup of contraction mappings of a finite chain. The Research Council of Oman Research grant proposal No. ORG/CBS/12/007.
[26] Umar, A. (1993). On the semigroups of partial one-to-one order-decreasing finite transformations, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, 123(2): 355–363.
[27] Umar A., Zubairu, M. M. (2018). Some Remarks about Semigroups of Partial Contraction Mappings of a Finite Chain. Conference of Proceeddings, North British Semigroups and Applications Network (NBSAN) 27th, University of York.
[28] Umar, A., Zubairu, M. M. (2021). On certain semigroup of contraction mappings of a finite chain. Algebra and Descrete Mathematics, 32(2): 299–320.
[2] Ali, B., Jada, M. A., Zubairu, M. M. (2023). Nilpotents in semigroups of order-preserving partial contraction mappings of a finite chain, Submitted.
[3] Al-Kharousi, F., Kehinde, R., Umar, A. (2014). Combinatorial results for certain semigroups of partial isometries of a finite chain. Australasian Journal of Combinatorics, 58(3): 365–375.
[4] Al-Kharousi, F., Kehinde, R., Umar, A. (2016). On the semigroup of partial isometries of a finite chain. Communications in Algebra, 44(2): 639–647.
[5] Ayik, G. Ayik, H. Howie, J. M., Ünlü, Y. (2008). Rank properties of semigroup of singular transformation on a finite set. Communications in Algebra, 36(7):2581–2587.
[6] Bugay, L. (2020). On the ranks of certain ideals of monotone contractions. Hacettepe Journal of Mathematics and Statistics, 49(6): 1988–1996
[7] Bugay, L., Melek, Y., Ayik, H. (2018). The Ranks Of Certain Semigroups of Partial Isometries. Semigroup Forum, 97: 214–222.
[8] Fountain, J. (1979). Adequate semigroups. Proceedings of the Edinburgh Mathematical Society, 22(2): 113–125.
[9] Fountain, J. (1982). Abundant semigroups. Proceedings of the London Mathematical Society, 3(1): 103–129.
[10] Garba, G. U. (1994). Nilpotents in semigroup of partial one-to-one order preserving mappings. Semigroup Forum, 48: 37–49.
[11] Garba, G. U. (1994). Nilpotents in semigroups of partial order-preserving transformations. Proceedings of the Edinburgh Mathematical Society, 37: 361–377.
[12] Garba, G. U. (1994). On the nilpotent ranks of certain semigroups of transformations, Glasgow Mathematical Journal, 36(1): 1–9.
[13] Ganyushkin, O., Mazorchuk, V. (2009). Classical Finite Transformation Semigroups. Springer-Verlag: London Limited.
[14] Gomes, G. M. S., Howie, J. M. (1987). Nilpotents in finite symmetric inverse semigroups, Proceedings of the Edinburgh Mathematical Society, 30: 383-395.
[15] Gomes, G. M. S., Howie, J. M. (1992). On the ranks of certain finite semigroups of transformations. Semigroup Forum, 45: 272–282.
[16] Green, J. A. (1951). On the structure of semigroups. Annals of Mathematics, 54(2):163–172.
[17] Howie, J. M., McFadden, R. B. (1990). Idempotent rank in finite full transformation semigroups, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, 114: 161-167.
[18] Howie, J. M. (1995). Fundamentals of semigroup theory. London Mathematical Society, New series 12. The Clarendon Press, Oxford University Press.
[19] Higgins, P. M. (1992). Techniques of semigroup theory. Oxford university Press.
[20] Imam, A. T., Usman, L., Idris, l., Ibrahim, S. (2024). Quasi-idempotent in finite semigroup of partial order-preserving transformation. International Journal of Mathematical Sciences and Optimization: Theory and Application, 10(1):53–59.
[21] Levi, I. (2006). Nilpotent Ranks of semigroups of partial transformations. Semigroup Forum, 72: 459–476
[22] Ladraji, A., Umar A. (2004). Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra, 278: 342–359.
[23] Peter, I. R., Mogbonju, M. M., Adeniji, A. O., Akinwunmi, S. A., Ibrahim, A. (2024). Some geometric characterization of star-like 3D conjugacy C3ωn∗ on partial one-one transformation semigroups. International Journal of Mathematical Sciences and Optimization: Theory and Application, 10(3):57–66.
[24] Toker K. (2020). Ranks of some subsemigroups of full contraction mappings on a finite chain. Journal of Balikesir University Institute of Science and Technology, 22(2): 403–414.
[25] Umar, A., Al-Kharousi, F. (2012). Studies in semigroup of contraction mappings of a finite chain. The Research Council of Oman Research grant proposal No. ORG/CBS/12/007.
[26] Umar, A. (1993). On the semigroups of partial one-to-one order-decreasing finite transformations, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, 123(2): 355–363.
[27] Umar A., Zubairu, M. M. (2018). Some Remarks about Semigroups of Partial Contraction Mappings of a Finite Chain. Conference of Proceeddings, North British Semigroups and Applications Network (NBSAN) 27th, University of York.
[28] Umar, A., Zubairu, M. M. (2021). On certain semigroup of contraction mappings of a finite chain. Algebra and Descrete Mathematics, 32(2): 299–320.
Published
2024-09-30
How to Cite
Ali , B., Jada , M. A., & Zubairu , M. M. (2024). On Rank of Semigroup of Order-Preserving Order-Decreasing Partial Contraction Mappings on a Finite Chain. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(4), 1 - 11. Retrieved from http://ijmso.unilag.edu.ng/article/view/2384
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