On the Dominions of Certain Semigroups of Transformations

Halima H. Assiri; Jehan A. Albar

paper

On the Dominions of Certain Semigroups of Transformations

arXiv:2606.28477

Abstract

In the full transformation semigroup $T_{n}$ on a finite chain $X_{n}$ , let $D_{n} = {α \in T_{n} : (\forall x \in X_{n}) xα \leq x}$ be the subsemigroup of all order-decreasing maps of $T_{n}$ , and let $O_{n} = {α \in T_{n} : (\forall x, y \in X_{n}) x \leq y \Rightarrow xα \leq y α}$ be the subsemigroup of all order-preserving maps of $T_{n}$ . The Catalan monoid $C_{n}$ is a semigroup of all order-decreasing and order-preserving full transformations of $X_{n}$ . In this paper, it is shown that $O_{n}$ is closed in $T_{n}$ . Also, the dominion of $D_{n}$ and the dominion of $C_{n}$ in $T_{n}$ , denoted by $D o m_{T_{n}} (D_{n})$ and $D o m_{T_{n}} (C_{n})$ , are characterized, and it is shown that they are regular idempotent-generated subsemigroups of $T_{n}$ . Moreover, a formula for the number of their elements and their idempotents is given.