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

Abstract

We attach to each $⟨0,\vee ⟩$-semilattice a graph ${\mathbf{G}}_{\mathbf{S}}$ whose vertices are join-irreducible elements of $\mathbf{S}$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph ${\mathbf{G}}_{\mathbf{S}}$ both when $\mathbf{S}$ is a join-semilattice and when it is a lattice. We call a $⟨0,\vee ⟩$-semilattice $\mathbf{S}$ particle provided that the set of its join-irreducible elements join-generates $\mathbf{S}$ 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.