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

## TL;DR

This paper explores the structure of join-semilattices and lattices through associated graphs, revealing that the congruence lattice of certain 'particle' lattices is anti-isomorphic to a lattice of hereditary subsets of these graphs.

## Contribution

It introduces the concept of particle lattices and characterizes their congruence lattices via graph-theoretic and topological methods, extending previous results.

## Key findings

- The graph $oldsymbol{G}_{oldsymbol{S}}$ encodes join-irreducible elements and their dependencies.
- The congruence lattice of a particle lattice is anti-isomorphic to hereditary subsets of its graph.
- This extends known results from principally chain finite lattices to a broader class.

## 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.