| Title:
|
The graphs of join-semilattices and the shape of congruence lattices of particle lattices (English) |
| Author:
|
Růžička, Pavel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
275-291 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\boldsymbol S$ 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. (English) |
| Keyword:
|
join-semilattice |
| Keyword:
|
lattice |
| Keyword:
|
join-irreducible |
| Keyword:
|
dependency |
| Keyword:
|
chain condition |
| Keyword:
|
particle |
| Keyword:
|
atomistic |
| Keyword:
|
congruence |
| MSC:
|
06A12 |
| MSC:
|
06A15 |
| MSC:
|
06B10 |
| MSC:
|
06F30 |
| idZBL:
|
Zbl 06837065 |
| idMR:
|
MR3708773 |
| DOI:
|
10.14712/1213-7243.2015.214 |
| . |
| Date available:
|
2017-11-22T09:18:26Z |
| Last updated:
|
2019-10-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146911 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
