Natural partial order and finiteness conditions on semigroups of linear transformations with invariant subspaces

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

“…Using this characterization, we prove that f ∈ ureg(L(V )) if and only if nullity(f ) = corank(f ) and then alternatively show that L(V ) is unit-regular if and only if V is finitedimensional. We also give a new and simple proof of [2,Theorem 11]. We first prove a list of key lemmas.…”

“…An element s of a semigroup S with identity is called unit-regular if there exists a unit u ∈ S such that sus = s. A unit-regular semigroup is a semigroup with identity in which every element is unit-regular. The notion of unit-regularity, which was introduced by Ehrlich [8] within the context of rings, has consecutively received wide attention from many semigroup theorists (see e.g., [1,2,3,7,9,10,18,19,27]). In 1980, Alarcao [7, Proposition 1] proved that a semigroup with identity is unitregular if and only if it is factorizable.…”

“…In [21,Proposition 3.1], the authors characterized regular elements in L(V, W ) and subsequently proved that L(V, W ) is regular if and only if W is trivial. Chaiya [2,Theorem 11] gave a necessary and sufficient condition for L(V, W ) to be unit-regular. Several other interesting properties of L(V, W ) have been studied (see, e.g., [2,4,13,22]).…”

“…Chaiya [2,Theorem 11] gave a necessary and sufficient condition for L(V, W ) to be unit-regular. Several other interesting properties of L(V, W ) have been studied (see, e.g., [2,4,13,22]).…”

Comments on “Semigroups of transformations whose restrictions belong to a given semigroup”

“…The semigroup L S(W ) (V ) is also a generalization of the subsemigroup L(V, W ) = {f ∈ L(V ) : W f ⊆ W } of L(V ), since L L(W ) (V ) = L(V, W ). The semigroup L(V, W ) has been studied by several authors (see, e.g., [1,8,12]).…”