On left restriction semigroups with zero

In this article, we give the notion of left restriction meet-semigroup, and establish some results regarding atomistic left restriction semigroups. Then we discuss decompositions of (non-zero) semigroups with zero by proving a decomposition theorem. We also show that every atomistic left restriction semigroup S can be decomposed as an orthogonal sum of atomistic left restriction semigroups Ni, where each summand Ni is an irreducible ideal of S. Finally, properties of the summands Ni, when S embeds in some PT X the partial transformation monoid on a set X, are investigated.