## Pavel Růžička

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

Comment.Math.Univ.Carolin. 58,3 (2017) 275-291.**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

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