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(2.3122n) and Ω(1.7250n), 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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Acknowledgements
The author wishes to thank the anonymous referee for helpful comments, and Peter Jipsen for suggesting the study of distributive lattices with the vertical 2-sum. Computational resources were provided by CSC – IT Center for Science and by the Aalto Science-IT project.
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Open access funding provided by Aalto University.
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Data availability statement
The data generated during this study are partially available in the EUDAT B2SHARE repository [19]. This contains the exhaustive listings of pieces and specials, up to sizes of 33 elements (modulars) and 55 elements (distributives). The data are stored in digraph6 format [11] and compressed with xz [20]. The larger lattices, modulars of 34–35 elements and distributives of 56–60 elements, are currently not available in EUDAT due to their size, but are available from the author on reasonable request.
Accompanying program code is available in Bitbucket [14]. This includes C programs to generate the pieces and the specials, to classify and count them by symmetry type, and to perform the Cartesian counting. Also included is SageMath code for verifying the exponential lower bounds.
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Kohonen, J. Cartesian Lattice Counting by the Vertical 2-sum. Order 39, 113–141 (2022). https://doi.org/10.1007/s11083-021-09569-0
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DOI: https://doi.org/10.1007/s11083-021-09569-0