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

Pavel Růžička

Commentationes Mathematicae Universitatis Carolinae (2017)

  • Volume: 58, Issue: 3, page 275-291
  • ISSN: 0010-2628

Abstract

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We attach to each 〈 0 , ∨ 〉 -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 〈 0 , ∨ 〉 -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.

How to cite

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Růžička, Pavel. "The graphs of join-semilattices and the shape of congruence lattices of particle lattices." Commentationes Mathematicae Universitatis Carolinae 58.3 (2017): 275-291. <http://eudml.org/doc/294133>.

@article{Růžička2017,
abstract = {We attach to each $\langle 0,\vee \rangle $-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 $\langle 0,\vee \rangle $-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.},
author = {Růžička, Pavel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {join-semilattice; lattice; join-irreducible; dependency; chain condition; particle; atomistic; congruence},
language = {eng},
number = {3},
pages = {275-291},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The graphs of join-semilattices and the shape of congruence lattices of particle lattices},
url = {http://eudml.org/doc/294133},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Růžička, Pavel
TI - The graphs of join-semilattices and the shape of congruence lattices of particle lattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 3
SP - 275
EP - 291
AB - We attach to each $\langle 0,\vee \rangle $-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 $\langle 0,\vee \rangle $-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.
LA - eng
KW - join-semilattice; lattice; join-irreducible; dependency; chain condition; particle; atomistic; congruence
UR - http://eudml.org/doc/294133
ER -

References

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