0. Introduction. On a semigroup 5 let the relation 01*, sometimes denoted by 01*-, be defined by x0l*y€> [(\/s,t In Section 2 we define the notions of k-semidirect product and full restricted semidirect product. These notions are not given in full generality but rather as they are needed here (first (second) component a semilattice (left type-v4 semigroup)).In Section 3 we prove that, given a left type-,/1 semigroup S and a left type-/4 congruence p on 5, satisfying p n 01* = i, the identity relation, then S is isomorphic to a well determined subsemigroup Tof a A-semidirect product A *^S/p, with A a semilattice.Moreover 0i*-c 0i%, )S/p .In particular, if S is proper, i.e. if a n 01* = t, where a is the least right cancellative congruence on 5, we obtain that 5 is isomorphic to an M-semigroup T, [2], which is embedded into a semidirect product of a semilattice by a right cancellative monoid. Regarding this we generalize Fountain's representation theorem for proper left type-^4 semigroups, as well as O'Carroll's embedding theorem for E-unitary inverse semigroups, [6].A related result, which states that each proper left type-/l semigroup is embeddable into a reverse semidirect product of a semilattice by a right cancellative monoid, was recently proven using the categorical approach in [4].In Section 4 we apply our methods to a certain class of left type-/l semigroups whose intersection with the class of inverse semigroups consists precisely of the £-reflexive ones, in the sense of [8].The notation and terminology of [8] will be used throughout the paper, whenever possible. In particular, for any congruence, s often denotes the congruence class containing s, and for H, K<=S, HK means the set product, i.e. HK = {hk\h e H, k e K}.