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 $\overline{T}(X,Y)=\{f\in T(X):Yf\subseteq Y\}$ and $\overline{L}(V,W)=\{f\in L(V):Wf\subseteq W\}$ 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 $\mathrm{nullity}(f)=\mathrm{corank}(f)$. 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.

arXiv comment15 pages