On isolated subsemigroups of order-decreasing transformation semigroups

Year 2026, Issue: Advanced Online Publication, 1 - 7

Melek Yağcı , Hayrullah Ayik

https://doi.org/10.15672/hujms.1632145

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] A.H. Clifford and G.B. Preston, The algebraic theory of semigroups. Vol. I., Amer. Math. Soc., Providence, 1961.