Let [Formula: see text] be the full transformation monoid on a nonempty set [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be semi-balanced if the collapse of [Formula: see text] is equal to the defect of [Formula: see text]. In this paper, we prove that an element of [Formula: see text] is unit-regular if and only if it is semi-balanced. For a partition [Formula: see text] of [Formula: see text], we characterize unit-regular elements in the monoid [Formula: see text] under composition. We characterize regular elements in the submonoids [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the equivalence induced by [Formula: see text]. We also characterize unit-regular elements in [Formula: see text], [Formula: see text], and the other two known submonoids of [Formula: see text].

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Let [Formula: see text] be a nonempty set and let [Formula: see text] be the full transformation semigroup on [Formula: see text]. The main objective of this paper is to study the subsemigroup [Formula: see text] of [Formula: see text] defined by [Formula: see text] where [Formula: see text] is a fixed nonempty subset of [Formula: see text]. We describe regular elements in [Formula: see text], and show that [Formula: see text] is regular if and only if [Formula: see text] is finite. We characterize unit-regular elements in [Formula: see text], and prove that [Formula: see text] is unit-regular if and only if [Formula: see text] is finite. We characterize Green’s relations on [Formula: see text], and prove that [Formula: see text] on [Formula: see text] if and only if [Formula: see text] is finite. We also determine ideals of [Formula: see text] and investigate its kernel. This paper extends several results that have appeared in the literature.

Let T (X) (resp. L(V)) be the semigroup of all transformations (resp. linear transformations) of a set X (resp. vector space V ). For a subset Y of X and a subsemigroup S(Y ) of T (Y ), consider the subsemigroupWe give a new characterization for T S(Y ) (X) to be a regular semigroup [inverse semigroup]. For a subspace W of V and a subsemigroup S(W ) of L(W ), we define an analogous subsemigroup L S(WWe describe regular elements in L S(W ) (V ) and determine when L S(W ) (V ) is a regular semigroup [inverse semigroup, completely regular semigroup]. If S(Y ) (resp. S(W )) contains the identity of T (Y ) (resp. L(W )), we describe unit-regular elements in T S(Y ) (X) (resp. L S(W ) (V )) and determine when T S(Y ) (X) (resp. L S(W ) (V )) is a unit-regular semigroup.L S(W ) (V ) = {f ∈ L(V ) : f ↾W ∈ S(W )},