Unit-regular elements in restrictive semigroups of transformations and linear operators
Mosarof Sarkar, Shubh N. Singh · arXiv (Cornell University) · 2021
Let $T(X)$ be the full transformation semigroup on a set $X$ and let $L(V)$ be the semigroup under composition of all linear operators on a vector space $V$ over a field. For a nonempty subset $Y$ of $X$ and a subspace $W$ of $V$, we consider the restrictive semigroups $\overline{T}(X, Y) = \{f\in T(X)\mid Yf \subseteq Y\}$ and $\overline{L}(V, W) = \{f\in L(V)\mid Wf \subseteq W\}$ under composition. We characterize unit-regular elements in $\overline{T}(X, Y)$ and $\overline{L}(V, W)$. Utilizing these, we characterize unit-regularity of $\overline{T}(X, Y)$ and $\overline{L}(V, W)$. We prove that $f\in L(V)$ is unit-regular if and only if nullity$(f) = {\rm corank}(f)$. A transformation semigroup is called semi-balanced if all its elements are semi-balanced. We determine a necessary and sufficient condition for $\overline{T}(X, Y)$ and $L(V)$ to be semi-balanced.