Title: The graphs of join-semilattices and the shape of congruence lattices of particle lattices,
Abstract: We attach to each $\langle 0, \vee \rangle$-semilattice a graph
$\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements
of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency
relation. We study properties of the graph $\boldsymbol{G}_{\boldsymbol{S}}$
both when $\boldsymbol{S}$ is a join-semilattice and when it is a lattice. We
call a $\langle 0, \vee \rangle$-semilattice $\boldsymbol{S}$ particle provided
that the set of its join-irreducible elements join-generates $\boldsymbol{S}$
and it satisfies DCC. We prove that the congruence lattice of a particle
lattice is anti-isomorphic to the lattice of hereditary subsets of the
corresponding graph that are closed in a certain zero-dimensional topology.
Thus we extend the result known for principally chain finite lattices.