Cartesian Lattice Counting by the Vertical 2-sum

Jukka Kohonen*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

A vertical 2-sum of a two-coatom lattice L and a two-atom lattice U is obtained by removing the top of L and the bottom of U, and identifying the coatoms of L with the atoms of U. This operation creates one or two nonisomorphic lattices depending on the symmetry case. Here the symmetry cases are analyzed, and a recurrence relation is presented that expresses the number of nonisomorphic vertical 2-sums in some desired family of graded lattices. Nonisomorphic, vertically indecomposable modular and distributive lattices are counted and classified up to 35 and 60 elements respectively. Asymptotically their numbers are shown to be at least omega(2.3122(n)) and omega(1.7250(n)), where n is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in n.

Original languageEnglish
Number of pages29
JournalORDER: A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS
DOIs
Publication statusE-pub ahead of print - 25 May 2021
MoE publication typeA1 Journal article-refereed

Keywords

  • Counting
  • Vertical 2-sum
  • Modular lattice
  • Distributive lattice

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