Unit-regular and semi-balanced elements in various semigroups of transformations

TL;DR

This paper characterizes unit-regular and semi-balanced elements in specific semigroups of transformations, providing conditions for their structure and regularity, especially in relation to linear transformations and subspace invariance.

Contribution

It offers a detailed description of unit-regular elements in transformation semigroups and establishes criteria for these semigroups to be unit-regular or semi-balanced.

Findings

Unit-regular elements in $ar{T}(X,Y)$ and $ar{L}(V,W)$ are characterized.

A linear transformation is unit-regular iff nullity equals corank.

The semigroup $L(V)$ is unit-regular iff $V$ is finite-dimensional.

Abstract

Let $T(X)$ be the full transformation semigroup on a set $X$, and let $L(V)$ be the semigroup under composition of all linear transformations on a vector space $V$ over a field. For a subset $Y$ of $X$ and a subspace $W$ of $V$, consider the semigroups $\overline{T}(X, Y) = \{f\in T(X)\colon Yf \subseteq Y\}$ and $\overline{L}(V, W) = \{f\in L(V)\colon Wf \subseteq W\}$ under composition. We describe unit-regular elements in $\overline{T}(X, Y)$ and $\overline{L}(V, W)$. Using these, we determine when $\overline{T}(X, Y)$ and $\overline{L}(V, W)$ are unit-regular. We prove that $f\in L(V)$ is unit-regular if and only if ${\rm nullity}(f) = {\rm corank}(f)$. We alternatively prove that $L(V)$ is unit-regular if and only if $V$ is finite-dimensional. A semi-balanced semigroup is a transformation semigroup whose all elements are semi-balanced. We give necessary and sufficient conditions for $\overline{T}(X, Y)$, $\overline{L}(V, W)$ and $L(V)$ to be semi-balanced.