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. Our results provide an alternative proof that $$f\in L(V)$$ is unit-regular if and only if $${{\,\mathrm{\text{ nullity }}\,}}(f) = {{\,\mathrm{\text{ corank }}\,}}(f)$$ , and 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.