The monoid of all order-decreasing full transformations

Abstract

Let ${\mathcal{D}}_n$ be the monoid of all order-decreasing full transformations on $\mathbf{n}=\{1,\dots ,n\}$. In this paper, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of ${\mathcal{D}}_n$. Moreover, we characterize the abundance of the maximal (idempotent generated) subsemigroups of ${\mathcal{D}}_n$. We also characterize the abundance of the ideals of ${\mathcal{D}}_n$.

KEYWORDS:

• Abundant
• left abundant
• maximal idempotent generated subsemigroup
• maximal subsemigroup
• right abundant

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M20
• 20M10

Acknowledgments

The authors would like to thank the referee for his/her valuable suggestions and comments which help to improve the presentation of this article.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No.12261022) and the National Natural Science Foundation of China (No.11461014).