On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range

TL;DR

This paper investigates the algebraic structure of semigroups of orientation-preserving transformations on infinite chains with restricted ranges, focusing on their relative ranks and minimal generating sets.

Contribution

It provides a calculation of the relative rank of the semigroup of orientation-preserving transformations modulo order-preserving transformations, including a characterization of minimal generating sets when the range equals the entire set.

Findings

Calculated the relative rank of $OP(X,Y)$ modulo $O(X,Y)$

Characterized minimal generating sets for $OP(X,Y)$ when $Y = X$

Extended understanding of the algebraic structure of transformation semigroups on infinite chains

Abstract

Let $X$ be an infinite linearly ordered set and let $Y$ be a nonempty subset of $X$. We calculate the relative rank of the semigroup $OP(X,Y)$ of all orientation-preserving transformations on $X$ with restricted range $Y$ modulo the semigroup $O(X,Y)$ of all order-preserving transformations on $X$ with restricted range $Y$. For $Y = X$, we characterize the relative generating sets of minimal size.