On the ranks of certain ideals of monotone contractions

Year 2020, Volume: 49 Issue: 6, 1988 - 1996, 08.12.2020

Leyla Bugay

https://doi.org/10.15672/hujms.604715

Abstract

Let $T_{n}$ be the (full) transformation semigroup, and let $OCT_{n}$ and $ORCT_{n}$ be its subsemigroups of isotone contractions and of monotone contractions on a finite chain $X_{n}=\{1,\ldots ,n\}$ under its natural order, respectively. In this study, we obtain the ranks of the ideals $OCT_{n,r}=\{\alpha\in OCT_{n}\,:\, |im(\alpha)|\leq r\}$ and $ORCT_{n,r}=\{ \alpha \in ORCT_{n}\,:\, |im(\alpha)|\leq r\}$ for $1\leq r\leq n-1$.

Keywords

isotone/antitone/monotone transformation, contraction, rank

References

• [1] A.D. Adeshola, Some semigroups of full contraction mappings of a finite chain, PH.D Thesis, University of Ilorin, Nigeria, 2013.
• [2] A.D. Adeshola and A. Umar, Combinatorial results for certain semigroups of orderpreserving full contraction mappings of a Finite Chain, J. Combin. Math. Combin. Comput. 106, 37–49, 2018.
• [3] H. Ayık and L. Bugay, Generating sets of finite transformation semigroups PK(n, r) and K(n, r). Comm. Algebra, 43, 412–422, 2015.
• [4] O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer-Verlag, Berlin, Germany, 2009.
• [5] G.U. Garba, Idempotents in partial transformation semigroups, Proc. Roy. Soc. Edinburgh, 116 (A), 359–366, 1990.
• [6] G.U. Garba, On the idempotent ranks of certain semigroups of order-preserving transformations, Port. Math. 51, 185–204, 1994.
• [7] G.U. Garba and M.J. Ibrahim and A.T. Imam, On certain semigroups of full contraction maps of a finite chain, Turkish J. Math. 41, 500–507, 2017.
• [8] G.M.S. Gomes and J.M. Howie, On the ranks of certain finite semigroups of transformations, Math. Proc. Cambridge Philos. Soc. 101 (3), 395–403, 1987.
• [9] G.M.S. Gomes and J.M. Howie, On the ranks of certain semigroups of orderpreserving transformations, Semigroup Forum, 45 (3), 272–282, 1992.
• [10] P.M. Higgins J.M. Howie, J.D. Mitchell and N. Ruskuc, Countable versus uncountable rank in finite semigroups of transformations and relations, Proc. Edinb. Math. Soc. 46, 531–544, 2003.
• [11] J.M. Howie, Idempotent generators in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh. Sect. A, 81 (3-4), 317–323, 1978.
• [12] J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, New York, USA, 1995.
• [13] J.M. Howie and R.B. McFadden, Idempotent rank in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh. Sect. A, 114 (3-4), 161–167, 1990.
• [14] G.R. Ibrahim, A.T. Imam, A.D. Adeshola and G.N. Bakare, Some algebraic properties of order-preserving full contraction transformation semigroup, J. Semigroup Theory Appl. 2019 (2), 2019.
• [15] K. Toker, Ranks of some subsemigroups of full contraction mappings on a finite chain (submitting).
• [16] R.J. Wilson and J.J. Watkins, Graphs, An Introductory Approach, A First Course in Discrete Mathematics, Jon Wiley & Sons Inc., Toronto, 1990.
• [17] P. Zhao and V.H. Fernandes, The Ranks of ideals in various transformation monoids, Comm. Algebra, 43, 674–692, 2015.