Slim lattice

In lattice theory, a mathematical discipline, a finite lattice is slim if no three join-irreducible elements form an antichain.[1] Every slim lattice is planar. A finite planar semimodular lattice is slim if and only if it contains no cover-preserving diamond sublattice M3 (this is the original definition of a slim lattice due to George Grätzer and Edward Knapp).[2]

Notes

  1. ^ Czédli & Schmidt (2012)
  2. ^ Grätzer & Knapp (2007)

References

  • Grätzer, George (2016). The congruences of a finite lattice. A "proof-by-picture" approach (2nd ed.). Cham, Switzerland: Birkhäuser/Springer. doi:10.1007/978-3-319-38798-7. ISBN 978-3-319-38796-3. MR 3495851.
  • Grätzer, George; Knapp, Edward (2007). "Notes on planar semimodular lattices. I. Construction". Acta Sci. Math. (Szeged). 73 (3–4): 445–462. arXiv:0705.3366. MR 2380059.
  • Czédli, Gábor; Schmidt, E. Tamás (2012). "Slim semimodular lattices. I. A visual approach" (PDF). Order. 29 (3): 481–497. doi:10.1007/s11083-011-9215-3. MR 2979644. S2CID 11481489.


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