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.
Categories: math.CO math.RA