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

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