| Title:
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Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices (English) |
| Author:
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Czédli, Gábor |
| Language:
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English |
| Journal:
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Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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149 |
| Issue:
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4 |
| Year:
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2024 |
| Pages:
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503-532 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_3$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset ${\rm J}({\rm Con} L)$ of join-irreducible congruences of $L$. We prove that if $1<n\in \mathbb N$ and $P$ is an $n$-element JConSPS-representable poset, then there exists a slim rectangular lattice $L$ such that ${\rm J}({\rm Con} L)\cong P$, the length of $L$ is at most $2n^2$, and $|L|\leq 4n^4$. This offers an algorithm to decide whether a finite poset $P$ is JConSPS-representable (or a finite distributive lattice is ``ConSPS-representable''). This algorithm is slow as G. Czédli, T. Dékány, G. Gyenizse, and J. Kulin proved in 2016 that there are asymptotically $\frac 12(k-2)! {\rm e}^2$ slim rectangular lattices of a given length $k$, where ${\rm e}$ is the famous constant $\approx 2.71828$. The known properties and constructions of JConSPS-representable posets can accelerate the algorithm; we present a new construction. (English) |
| Keyword:
|
slim rectangular lattice |
| Keyword:
|
slim semimodular lattice |
| Keyword:
|
planar semimodular lattice |
| Keyword:
|
congruence lattice |
| Keyword:
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lattice congruence |
| Keyword:
|
lamp |
| Keyword:
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$\mathcal C_1$-diagram |
| MSC:
|
06C10 |
| DOI:
|
10.21136/MB.2024.0006-23 |
| . |
| Date available:
|
2024-12-13T19:05:27Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152677 |
| . |
| 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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| Reference:
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| Reference:
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