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

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.