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 Con_c A, 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. Received December 10, 2004; accepted in final form June 6, 2005.