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Generating all finite modular lattices of a given size

Abstract

Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold [8] developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 24, semimodular lattices up to size 22, and lattices of size 19. We also show that 2n−3 is a lower bound for the number of nonisomorphic modular lattices of size n.

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Correspondence to Peter Jipsen.

Additional information

Dedicated to Brian Davey on the occasion of his 65th birthday

Presented by R. Quackenbush.

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Jipsen, P., Lawless, N. Generating all finite modular lattices of a given size. Algebra Univers. 74, 253–264 (2015). https://doi.org/10.1007/s00012-015-0348-x

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2010 Mathematics Subject Classification

  • Primary: 06C05
  • Secondary: 06C10
  • 05A15

Key words and phrases

  • modular lattices
  • semimodular lattices
  • counting up to isomorphism
  • orderly algorithm