Generating Modular Lattices of up to 30 Elements

Jukka Kohonen*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)

Abstract

An algorithm is presented for generating finite modular, semimodular, graded, and geometric lattices up to isomorphism. Isomorphic copies are avoided using a combination of the general-purpose graph-isomorphism tool nauty and some optimizations that handle simple cases directly. For modular and semimodular lattices, the algorithm prunes the search tree much earlier than the method of Jipsen and Lawless, leading to a speedup of several orders of magnitude. With this new algorithm modular lattices are counted up to 30 elements, semimodular lattices up to 25 elements, graded lattices up to 21 elements, and geometric lattices up to 34 elements. Some statistics are also provided on the typical shape of small lattices of these types.

Original languageEnglish
Pages (from-to)423–435
JournalOrder
Volume36
Issue number3
Early online date1 Jan 2018
DOIs
Publication statusPublished - Nov 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • Counting algorithm
  • Geometric lattices
  • Graded lattices
  • Modular lattices
  • Semimodular lattices

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