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

Abstract

We attach to each $\langle 0,\vee \rangle$-semilattice $\boldsymbol S$ 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 satisfies DCC and join-generates $\boldsymbol S$. We prove that the congruence lattice of a particle lattice is anti-isomorphic to the lattice of all 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.