Gröbner bases and the number of Latin squares related to autotopisms of order ≤7

Latin squares can be seen as multiplication tables of quasigroups, which are, in general, non-commutative and non-associative algebraic structures. The number of Latin squares having a fixed isotopism in their autotopism group is at the moment an open problem. In this paper, we use Gröbner bases to...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 42; no. 11; pp. 1142 - 1154
Main Authors: Falcón, R.M., Martín-Morales, J.
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.11.2007
Subjects:
Autotopism group
Gröbner basis
Latin square
Gröbner basis
Autotopism group
Latin square
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary:Latin squares can be seen as multiplication tables of quasigroups, which are, in general, non-commutative and non-associative algebraic structures. The number of Latin squares having a fixed isotopism in their autotopism group is at the moment an open problem. In this paper, we use Gröbner bases to describe an algorithm that allows one to obtain the previous number. Specifically, this algorithm is implemented in Singular to obtain the number of Latin squares related to any autotopism of Latin squares of order up to 7.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2007.07.004