Characterize the congruence lattices of slim, planar, semimodular (SPS) lattices
Statement
A finite lattice is SPS if it is slim (no $M_3$ sublattice; equivalently its poset of join-irreducibles has no $3$-element antichain), planar, and semimodular. Characterize the finite distributive lattices $D$ that arise as $\mathrm{Con}\,L$ for some SPS lattice $L$; equivalently, characterize the finite ordered sets $P$ that arise as $J(\mathrm{Con}\,L)$, the poset of join-irreducible congruences of an SPS lattice.
Acceptance. ADVANCES: a decision (with certificate) of representability for finite ordered sets $P$ beyond those currently settled — either an SPS lattice $L$ with $J(\mathrm{Con}\,L)\cong P$ (certificate: the lattice $L$), or a proof that a given small $P$ is not representable; or a new necessary property ruling out a previously-open family. FULLY RESOLVES: a checkable characterization of the finite $P$ arising as $J(\mathrm{Con}\,L)$ for SPS $L$. Provide the lattice/obstruction plus a verification script.
Background
G. Grätzer, 'The Congruences of a Finite Lattice: A Proof-by-Picture Approach' (2016) and 'An Open Problem on Congruences of Finite Lattices in Pictures' (arXiv:2303.00699); 'Notes on planar semimodular lattices VIII: Congruence lattices of SPS lattices' (arXiv:2104.13843). SPS lattices were introduced by Grätzer–Knapp (2007); more than four dozen papers have studied their congruences — this is an ACTIVE area, not a neglected one. Necessary conditions are known and strengthening: beyond distributivity, congruence lattices of SPS lattices satisfy at least seven published properties — the Two-Cover Condition ($P$ has no cover-preserving copy of a specific $5$-element poset $F_3$; Grätzer 2016/2020), four properties of Czédli (2021, incl. '3P3C'), and a joint Grätzer–Czédli property (2022); 'lamp' techniques give infinitely many further necessary properties and an exponential-time decision algorithm for finite instances. None is known to be sufficient — e.g. the $3$-element chain $C_3$ satisfies the Two-Cover Condition yet is provably NOT $J(\mathrm{Con}\,L)$ for any SPS $L$. A full characterization is missing. An attacker must bring: a finite enumeration engine — for small ordered sets $P$, algorithmically decide representability by building candidate SPS lattices via the rectangular/patch construction and computing their congruence posets; the resulting data (which small $P$ are/aren't representable) sharpens or refutes conjectured characterizations, and every instance is a finite certificate.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Grätzer, An Open Problem on Congruences of Finite Lattices in Pictures (2023) | arxiv |
| REF-02 | Notes on planar semimodular lattices VIII: Congruence lattices of SPS lattices | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.