Cartesian Lattice Counting by the Vertical 2-sum

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 a...

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Bibliographic Details
Published in:Order (Dordrecht) 2022-04, Vol.39 (1), p.113-141
Main Author: Kohonen, Jukka
Format: Article
Language:English
Subjects:
Algebra
Cartesian coordinates
Discrete Mathematics
Lattice modules
Lattices
Lattices (mathematics)
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Symmetry
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Summary: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 .
ISSN:0167-8094
1572-9273
DOI:10.1007/s11083-021-09569-0