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We attach to each $\langle 0, \vee \rangle$-semilattice a graph$\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elementsof $\boldsymbol{S}$ and whose edges correspond to the reflexive dependencyrelation. We study properties of the graph $\boldsymbol{G}_{\boldsymbol{S}}$both when $\boldsymbol{S}$ is a join-semilattice and when it is a lattice. Wecall a $\langle 0, \vee \rangle$-semilattice $\boldsymbol{S}$ particle providedthat the set of its join-irreducible elements join-generates $\boldsymbol{S}$and it satisfies DCC. We prove that the congruence lattice of a particlelattice is anti-isomorphic to the lattice of hereditary subsets of thecorresponding graph that are closed in a certain zero-dimensional topology.Thus we extend the result known for principally chain finite lattices.

The graphs of join-semilattices and the shape of congruence lattices of particle lattices