2021/06/15 by Sarkar, Mosarof, Singh, Shubh N.
#15A03 #15A04 #20M17 #20M20 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2106.08063
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 T(X, Y) = \f∈ T(X)\colon Yf ⊆ Y\ and L(V, W) = \f∈ L(V)\colon Wf ⊆ W\ under composition. We describe unit-regular elements in T(X, Y) and L(V, W). Using these, we determine when T(X, Y) and L(V, W) are unit-regular. We prove that f∈ 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 T(X, Y), L(V, W) and L(V) to be semi-balanced.