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 $\boldsymbol S$ \emph{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.
Keywords: join-semilattice; lattice; join-irreducible; dependency; chain condition; particle; atomistic; congruence
DOI: DOI 10.14712/1213-7243.2015.214
AMS Subject Classification: 06A12 06A15 06B10 06F30