On isolated subsemigroups of order-decreasing transformation semigroups

Abstract

For $n\in \mathbb{N}$, let $\mathcal{D}_{n}$ be the semigroup of all order-decrasing transformations on $X_{n}=\{1,\ldots ,n\}$, under its natural order. In this paper, we determine isolated, completely isolated, and (left/right) convex subsemigroups of $\mathcal{D}_{n}$. Furthermore, for $\{ 1\}\neq U\subset X_{n}$ which contains $1$, we find the rank of $\mathcal{D}_{n}[U] =\{ \alpha \in \mathcal{D}_{n} :U\subseteq X_{n} \}$ which is a convex subsemigroup of $\mathcal{D}_{n}$.

Keywords

• order-decreasing transformation
• (completely) isolated subsemigroup
• convex subsemigroup
• generating set
• rank

References

1. [1] A.H. Clifford and G.B. Preston, The algebraic theory of semigroups. Vol. I., Amer. Math. Soc., Providence, 1961.
2. [2] A.H. Clifford and G.B. Preston, The algebraic theory of semigroups. Vol. II., Amer. Math. Soc., Providence, 1967.
3. [3] O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer-Verlag, London, 2009.
4. [4] J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, New York, 1995.
5. [5] E. Korkmaz and H. Ayık, Isolated subsemigroups of order-preserving and decreasing transformation semigroups, Bull. Malays. Math. Sci. Soc. 45, 663-675, 2022.
6. [6] A. Laradji and A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum 69, 184-200, 2004.
7. [7] V. Mazorchuk and G. Tsyaputa, Isolated subsemigroups in the variants of Tn, Acta Math. Univ. Comenianae 77(1), 63-84, 2008.
8. [8] A. Stronska, Nilpotent subsemigroups of a semigroup of order-decreasing transformations of a rooted tree, Algebra Discrete Math. 5(4), 126-140, 2007.

9. [9] L. Sun, A natural partial order on partition order-decreasing transformation semigroups, Bull. Iranian Math. Soc. 46, 1357-1369, 2020.
10. [10] G. Tsyaputa, Isolated and nilpotent subsemigroups in the variants of ISn, Algebra and Discrete Mathematics 5(1), 89-97, 2006.
11. [11] A. Umar, On the semigroups of order-decreasing finite full transformations, Proceedings of the Royal Society of Edinburg 120A, 129-142, 1992.
12. [12] A. Umar, Semigroups of Order-Decreasing Transformations, Ph.D. Thesis, University of St Andrews, 1992.
13. [13] A. Umar, Semigroups of order-decreasing transformations: the isomorphism theorem, Semigroup Forum 53, 220-224, 1996.
14. [14] M. Yagcı, On nilpotent subsemigroups of the order-decreasing transformation semigroups, Bull. Malays. Math. Sci. Soc. 46, article no. 53, 2023.
15. [15] H. Yang, and X. Yang, Automorphisms of partition order-decreasing transformation monoids, Semigroup Forum 85(3), 513-524, 2012.