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

Abstract

We attach to each -semilattice S a graph G(S) whose vertices are join-irreducible elements of S and whose edges correspond to the reflexive dependency relation. We study properties of the graph G(S) both when S is a join-semilattice and when it is a lattice.

We call a -semilattice S particle provided that the set of its join-irreducible elements satisfies DCC and join-generates 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.

Keywords

• join-semilattice
• lattice
• join-irreducible
• dependency
• chain condition
• particle
• atomistic
• congruence