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

TL;DR

This paper proves that for any finite poset with a certain representability property, there exists a slim rectangular lattice with bounded length and size that represents it, and provides an algorithm to determine such representability.

Contribution

It establishes bounds on the size and length of slim rectangular lattices representing JConSPS-posets and introduces a new construction to improve the representation algorithm.

Findings

Bound on lattice length: at most 2n^2.

Bound on lattice size: at most 4n^4.

Algorithm for JConSPS-representability decision.

Abstract

Following G. Gr\"atzer and E. Knapp (2007), a slim 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 J(Con $L$) of join-irreducible congruences of $L$. We prove that if $1<n\in N$ and $P$ is an $n$-element JConSPS-representable poset, then there exists a slim rectangular lattice $L$ such that J(Con $L$) is isomorphic to $P$, 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. Cz\'edli, T. D\'ek\'any, G. Gyenizse, and J. Kulin proved in 2016 that there are asymptotically $(k-2)!\cdot e^2/2$ many slim rectangular lattices of a given length $k$, where $e$ 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.