On relative ranks of finite transformation semigroups with restricted range

TL;DR

This paper determines the minimal generating sets for certain finite transformation semigroups with restricted ranges, focusing on their relative ranks modulo orientation-preserving and order-preserving transformations.

Contribution

It provides explicit calculations of the relative ranks and characterizes minimal generating sets for these semigroups, extending understanding of their algebraic structure.

Findings

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

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

Characterized minimal relative generating sets

Abstract

In this paper, we determine the relative rank of the semigroup $T(X,Y)$ of all transformations on a finite chain $X$ with restricted range $Y \subseteq X$ modulo the set $OP(X,Y)$ of all orientation-preserving transformation in $T(X,Y)$. Moreover, we state the relative rank of the semigroup $OP(X,Y)$ modulo the set $O(X,Y)$ of all order-preserving transformations in $OP(X,Y)$. In both cases we characterize the minimal relative generating sets.