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Local separation in distributive semilattices. (English) Zbl 1086.06003

Summary: We introduce the Local Separation Property (LSP) for distributive semilattices. We show that LSP holds in many semilattices of the form \(\text{Con}_c A\), 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.

MSC:

06A12 Semilattices
06B10 Lattice ideals, congruence relations
06D05 Structure and representation theory of distributive lattices
54H10 Topological representations of algebraic systems
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