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

@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 -