Generating Modular Lattices of up to 30 Elements

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Generating Modular Lattices of up to 30 Elements. / Kohonen, Jukka.

In: Order, Vol. 36, No. 3, 11.2019, p. 423–435.

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Kohonen, Jukka. / Generating Modular Lattices of up to 30 Elements. In: Order. 2019 ; Vol. 36, No. 3. pp. 423–435.

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@article{efbae9e1294d4a44be6645d9f3acbd7d,
title = "Generating Modular Lattices of up to 30 Elements",
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.",
keywords = "Counting algorithm, Geometric lattices, Graded lattices, Modular lattices, Semimodular lattices",
author = "Jukka Kohonen",
year = "2019",
month = "11",
doi = "10.1007/s11083-018-9475-2",
language = "English",
volume = "36",
pages = "423–435",
journal = "ORDER: A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS",
issn = "0167-8094",
publisher = "Springer Netherlands",
number = "3",

}

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TY - JOUR

T1 - Generating Modular Lattices of up to 30 Elements

AU - Kohonen, Jukka

PY - 2019/11

Y1 - 2019/11

N2 - 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.

AB - 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.

KW - Counting algorithm

KW - Geometric lattices

KW - Graded lattices

KW - Modular lattices

KW - Semimodular lattices

UR - http://www.scopus.com/inward/record.url?scp=85054333961&partnerID=8YFLogxK

U2 - 10.1007/s11083-018-9475-2

DO - 10.1007/s11083-018-9475-2

M3 - Article

VL - 36

SP - 423

EP - 435

JO - ORDER: A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS

JF - ORDER: A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS

SN - 0167-8094

IS - 3

ER -

ID: 28770605