Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices
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 comple...
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Institute of Mathematics of the Czech Academy of Science
2024-12-01
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| Series: | Mathematica Bohemica |
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| Online Access: | https://mb.math.cas.cz/full/149/4/mb149_4_4.pdf |
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| author | Gábor Czédli |
| author_facet | Gábor Czédli |
| author_sort | Gábor Czédli |
| collection | DOAJ |
| description | 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 $|L|\leq4n^4$. 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 $\frac12(k-2)! {\rm e}^2$ slim rectangular lattices of a given length $k$, where ${\rm e}$ is the famous constant $\approx2.71828$. The known properties and constructions of JConSPS-representable posets can accelerate the algorithm; we present a new construction. |
| format | Article |
| id | doaj-art-ea0bd8abfd7d4a928f9e640f05352296 |
| institution | OA Journals |
| issn | 0862-7959 2464-7136 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj-art-ea0bd8abfd7d4a928f9e640f053522962025-08-20T01:59:26ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362024-12-01149450353210.21136/MB.2024.0006-23MB.2024.0006-23Reducing the lengths of slim planar semimodular lattices without changing their congruence latticesGábor CzédliFollowing 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 $|L|\leq4n^4$. 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 $\frac12(k-2)! {\rm e}^2$ slim rectangular lattices of a given length $k$, where ${\rm e}$ is the famous constant $\approx2.71828$. The known properties and constructions of JConSPS-representable posets can accelerate the algorithm; we present a new construction.https://mb.math.cas.cz/full/149/4/mb149_4_4.pdf slim rectangular lattice slim semimodular lattice planar semimodular lattice congruence lattice lattice congruence lamp $\mathcal c_1$-diagram |
| spellingShingle | Gábor Czédli Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices Mathematica Bohemica slim rectangular lattice slim semimodular lattice planar semimodular lattice congruence lattice lattice congruence lamp $\mathcal c_1$-diagram |
| title | Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices |
| title_full | Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices |
| title_fullStr | Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices |
| title_full_unstemmed | Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices |
| title_short | Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices |
| title_sort | reducing the lengths of slim planar semimodular lattices without changing their congruence lattices |
| topic | slim rectangular lattice slim semimodular lattice planar semimodular lattice congruence lattice lattice congruence lamp $\mathcal c_1$-diagram |
| url | https://mb.math.cas.cz/full/149/4/mb149_4_4.pdf |
| work_keys_str_mv | AT gaborczedli reducingthelengthsofslimplanarsemimodularlatticeswithoutchangingtheircongruencelattices |