Cartesian Lattice Counting by the Vertical 2-sum

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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2022-04

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Department of Mathematics and Systems Analysis

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Language

en

Pages

29
113–141

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ORDER: A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS, Volume 39, issue 1

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 Ω(2.3122 n) and Ω(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.

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Keywords

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

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DOI

10.1007/s11083-021-09569-0

Citation

Kohonen, J 2022, ' Cartesian Lattice Counting by the Vertical 2-sum ', ORDER: A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS, vol. 39, no. 1, pp. 113–141 . https://doi.org/10.1007/s11083-021-09569-0

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https://urn.fi/URN:NBN:fi:aalto-202212076764

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[article-cris] Perustieteiden korkeakoulu / SCI