On the Relative Rank of Orientation-preserving or Orientation-reversing Transformation Semigroups with Restricted Range

Annual Meeting in Mathematics 2023

Abstract

Let $G$ and $U$ be subsets of a semigroup $S$. The rank of a semigroup $S$ is the minimal size of a generating set of $S$. By the definition of rank, it gives a new idea of definition of rank which is called the relative rank of $S$ modulo $U$ which is the minimal size of a subset $G$ such that $G\cup U$ is a generating set of $S$. A set $G$ is called a generating set of $S$ modulo $U$. Let $X$ be a finite chain and let $Y$ be a subchain of $X$. The semigroup $\mathcal{T}(X,Y)$ is so-called the full transformation semigroup on $X$ with restricted range $Y$ which is a subsemigroup of the semigroup $\mathcal{T}(X)$. In this work, we determine the relative rank of the semigroup $\mathcal{OPR}(X,Y)$ of all orientation-preserving or orientation-reversing transformations with restricted range modulo the semigroup $\mathcal{OD}(X,Y)$ of all order-preserving or order-reversing transformations with restricted range.