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Keywords:
slim rectangular lattice; slim semimodular lattice; planar semimodular lattice; congruence lattice; lattice congruence; lamp; $\mathcal C_1$-diagram
slim rectangular lattice; slim semimodular lattice; planar semimodular lattice; congruence lattice; lattice congruence; lamp; $\mathcal C_1$-diagram
Summary:
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.
References:
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