Local separation in distributive semilattices

We introduce the Local Separation Property (LSP) for distributive semilattices. We show that LSP holds in many semilattices of the form ConcA, where A is a lattice. On the other hand, we construct an abstract example of a distributive lattice without LSP. Our research is connected with the well known open problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of some lattice.