Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Czédli, Gábor. "Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices." Mathematica Bohemica 149.4 (2024): 503-532. <http://eudml.org/doc/299645>.

@article{Czédli2024,
abstract = {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 $|L|\le 4n^4$. 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 $\frac\{1\}\{2\}(k-2)! \{\rm e\}^2$ 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.},
author = {Czédli, Gábor},
journal = {Mathematica Bohemica},
keywords = {slim rectangular lattice; slim semimodular lattice; planar semimodular lattice; congruence lattice; lattice congruence; lamp; $\mathcal \{C\}_1$-diagram},
language = {eng},
number = {4},
pages = {503-532},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices},
url = {http://eudml.org/doc/299645},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Czédli, Gábor
TI - Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 4
SP - 503
EP - 532
AB - 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 $|L|\le 4n^4$. 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 $\frac{1}{2}(k-2)! {\rm e}^2$ 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.
LA - eng
KW - slim rectangular lattice; slim semimodular lattice; planar semimodular lattice; congruence lattice; lattice congruence; lamp; $\mathcal {C}_1$-diagram
UR - http://eudml.org/doc/299645
ER -