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

Commentationes Mathematicae Universitatis Carolinae (2017)

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

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topRůžička, 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 -

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