The Monoid Rank and Monoid Presentation of Order-Preserving and Order-Decreasing Full Contraction Mappings

Year 2021, Volume: 9 Issue: 4, 170 - 175, 31.12.2021

Kemal Toker

https://doi.org/10.36753/mathenot.807993

Cited By: 1

Abstract

Let $n \in \mathbb{Z}^{+}$ and $X_{n}=\{1,2,\ldots,n\}$ be a finite set. Let $\mathcal ODCT_{n}$ be the order-preserving and order-decreasing full contraction mappings on $X_{n}$. It is well known that $\mathcal ODCT_{n}$ is a monoid. In this paper, we have found the monoid rank and monoid presentation of $\mathcal ODCT_{n}$. In particular, we have proved that monoid rank of $\mathcal ODCT_{n}$ is $n-1$ for $n \in \mathbb{Z}^{+}$ and $$ is a monoid presentation of $\mathcal ODCT_{n}$ for $n \geq 3$.

Keywords

Contraction mappings, Generating set, Monoid Presentation

References

• [1] Adeshola, A.D., Umar, A.: Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain. Journal of Combinatorial Mathematics and Combinatorial Computing. 106, 37-49 (2018).
• [2] Ayık, G., Ayık H., Koç, M.: Combinatorial results for order-preserving and order-decreasing transformations. Turkish Journal of Mathematics. 35 (4), 617-625 (2011).
• [3] Ayık, H.: Presentations and efficiency of semigroups. Ph. D. Thesis. Universtiy of St Andrews (1998).
• [4] Garba, G.U., Ibrahim, M.J., Imam, A.T.: On certain semigroups of full contraction maps of a finite chain. Turkish Journal of Mathematics. 41 (3), 500-507 (2017).
• [5] Gomes,M.S.,Howie,J.M.:Ontheranksofcertainsemigroupsoforder-preservingtransformations.SemigroupForum. 45 (1), 272-282 (1992).
• [6] Higgins, P.M.: Combinatorial results for semigroups of order-preserving mappings. Mathematical Proceedings of the Cambridge Philosophical Society. 113 (2), 281-296 (1993).
• [7] Howie, J.M.: Fundamentals of semigroup theory. Oxford University Press. New York (1995).
• [8] Ruskuc, N.: Semigroup presentations. Ph. D. Thesis. Universtiy of St Andrews (1995).