Combinatorial Results on Some Nilpotent Subsemigroups of a Semigroup of Order-Decreasing Full Transformations

Abstract

Let $\mathcal{D}_{n}$ be the semigroup of all order-decreasing full transformations on $X_{n}=\{1,2,\ldots ,n\}$ under its natural order, and let $N(\mathcal{D}_{n})$ be the subsemigroup of all nilpotent elements of $\mathcal{D}_{n}$, where $n\in \mathbb{Z}^+$, the set of all positive integers. In this paper, for $1\leq r\leq n-1$, we determine the cardinality and rank of nilpotent subsemigroup $N(\mathcal{D}_{n,r}) =\{ \alpha\in N(\mathcal{D}_{n}) : \lvert\text{im}(\alpha)\rvert \leq r\}$ of $N(\mathcal{D}_{n})$. We then find the cardinalities of $\mathcal{D}_{n}^{2,2}$ and $N(\mathcal{D}_{n})^{p,p}$. Furthermore, we presents an alternative combinatorial approach to determine the cardinality and rank of ${\mathcal{D}}_{n}(\xi)=\{\alpha \in {\mathcal{D}}_{n}:\alpha ^k=\xi, \text{ for some } k\in {\mathbb {Z}}^{+}\}$, for all idempotent $\xi\in \mathcal{D}_{n}$ within the scope of this study. Here, for all $\alpha\in\mathcal{D}_{n}$, $\text{im}^{c}(\alpha)=\{t\in \text{im}(\alpha): \lvert t\alpha^{-1}\rvert\geq 2\}$. Besides, for all $2\leq p\leq r\leq n$ and $\mathcal{C}\in \{N(\mathcal{D}_n),\mathcal{D}_n\}$, $\mathcal{C}^{p}=\left\{\alpha\in \mathcal{C}: t\in \text{im}^{c}(\alpha) \text{ and } \lvert t\alpha^{-1}\rvert=p\right\}$ and $\mathcal{C}^{p,r}=\left\{\alpha\in \mathcal{C}^p:\bigg\lvert \bigcup\limits_{t\in \text{im}^{c}(\alpha)} t\alpha^{-1}\bigg\rvert=r\right\}$.

Keywords

• Order-decreasing
• transformations
• collapse
• nilpotent
• rank

References

1. O. Ganyushkin, V. Mazorchuk, Classical finite transformation semigroups, Springer-Verlag, 2009.
2. J. M. Howie, R. B. McFadden, Idempotent rank in finite full transformation semigroups, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 114 (3-4) (1990) 161-167.
3. N. Ruškuc, On the rank of completely 0-simple semigroups, Mathematical Proceedings of the Cambridge Philosophical Society 116 (2) (1994) 325-338.
4. E. Korkmaz, 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) (2022) 769-777.
5. A. Laradji, A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum 69 (2) (2004) 184-200.
6. A. Umar, On the semigroups of order-decreasing finite full transformations, Proceedings of the Royal Society of Edinburgh Section A Mathematics 120 (1-2) (1992) 129-142.
7. A. Umar, Semigroups of order-decreasing transformations: The isomorphism theorem, Semigroup Forum 53 (1) (1996) 220-224.
8. A. Umar, On the ranks of certain finite semigroups of order-decreasing transformations, Portugaliae Mathematica 53 (1) (1996) 23-32.

9. A. Umar, A class of (0)-idempotent-free transformation semigroups, Semigroup Forum 59 (1999) 74-78.
10. P. M. Higgins, Combinatorial results for semigroups of order-preserving mappings, Mathematical Proceedings of the Cambridge Philosophical Society 113 (2) (1993) 281-296.
11. E. Korkmaz, H. Ayık, Ranks of nilpotent subsemigroups of order-preserving and decreasing transformation semigroups, Turkish Journal of Mathematics 45 (4) (2021) 1626-1634.
12. M. Yağcı, E. Korkmaz, On nilpotent subsemigroups of the order-preserving and decreasing transformation semigroups, Semigroup Forum 101 (2) (2020) 486-496.
13. J. M. Howie, Fundamentals of semigroup theory, Oxford University Press, 1995.
14. M. Yağcı, On nilpotent subsemigroups of the order-decreasing transformation semigroups, Bulletin of the Malaysian Mathematical Sciences Society 46 (2) (2023) 53.
15. D. McAlister, Ash's theorem, finite nilpotent semigroups, and one-dimensional tiling semigroups, Manuscript of talk given at CAUL, October, 2001 and Northern Illinois University, July, (2002).