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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| Veröffentlicht in: | Journal of symbolic computation Jg. 42; H. 11; S. 1142 - 1154 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Falcón, R.M. Martín-Morales, J. |
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| Cites_doi | 10.1016/j.jsc.2005.09.007 10.1090/S0002-9947-1943-0009962-7 10.1090/S0002-9947-1944-0009963-X 10.1002/jcd.20105 |
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| Keywords | Gröbner basis Autotopism group Latin square |
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| References | Albert (b2) 1943; 54 Falcón, R.M., 2006. Latin squares associated to principal autotopisms of long cycles. Application in Cryptography. In: Proceedings of Transgressive Computing 2006: A Conference in Honor of Jean Della Dora. Granada. pp. 213–230 Bruck (b4) 1944; 55 Falcón, R.M., 2007. Cycle structures of autotopisms of the Latin squares of order up to 11. Ars Combinatoria (in press). Buchberger, B., 1965. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph. D. Thesis. University of Innsbruck. English translation 2006: An algorithm for finding the basis elements in the residue class ring modulo a zero dimensional polynomial ideal. (Logic, mathematics, and computer science: Interactions). Journal of Symbolic Computation. 41 (3–4), 475–511 (special issue) Laywine, Mullen (b11) 1998 Greuel, G.-M., Pfister, G., Schönemann, H., 2005. Adams, Loustaunau (b1) 1994; vol. 3 McKay, Meynert, Myrvold (b13) 2007; 15 Gago-Vargas, Hartillo-Hermoso, Martín-Morales, Ucha-Enríquez (b9) 2006; vol. 4194 Bayer, D., 1982. The division algorithm and the Hilbert scheme. Ph. D. Thesis. Harvard University Cox, Little, O’Shea (b6) 1997 3.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserlautern. Martín-Morales, J., 2006. Sudoku and Gröbner bases. In: Proceedings of Transgressive Computing 2006: A Conference in Honor of Jean Della Dora. Granada. pp. 303–310 Bruck (10.1016/j.jsc.2007.07.004_b4) 1944; 55 McKay (10.1016/j.jsc.2007.07.004_b13) 2007; 15 Gago-Vargas (10.1016/j.jsc.2007.07.004_b9) 2006; vol. 4194 10.1016/j.jsc.2007.07.004_b3 10.1016/j.jsc.2007.07.004_b5 Albert (10.1016/j.jsc.2007.07.004_b2) 1943; 54 10.1016/j.jsc.2007.07.004_b8 10.1016/j.jsc.2007.07.004_b7 Cox (10.1016/j.jsc.2007.07.004_b6) 1997 10.1016/j.jsc.2007.07.004_b12 Adams (10.1016/j.jsc.2007.07.004_b1) 1994; vol. 3 10.1016/j.jsc.2007.07.004_b10 Laywine (10.1016/j.jsc.2007.07.004_b11) 1998 |
| References_xml | – reference: Martín-Morales, J., 2006. Sudoku and Gröbner bases. In: Proceedings of Transgressive Computing 2006: A Conference in Honor of Jean Della Dora. Granada. pp. 303–310 – reference: Falcón, R.M., 2007. Cycle structures of autotopisms of the Latin squares of order up to 11. Ars Combinatoria (in press). – reference: Buchberger, B., 1965. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph. D. Thesis. University of Innsbruck. English translation 2006: An algorithm for finding the basis elements in the residue class ring modulo a zero dimensional polynomial ideal. (Logic, mathematics, and computer science: Interactions). Journal of Symbolic Computation. 41 (3–4), 475–511 (special issue) – reference: Falcón, R.M., 2006. Latin squares associated to principal autotopisms of long cycles. Application in Cryptography. In: Proceedings of Transgressive Computing 2006: A Conference in Honor of Jean Della Dora. Granada. pp. 213–230 – volume: 15 start-page: 98 year: 2007 end-page: 119 ident: b13 article-title: Small Latin squares, quasigroups and loops publication-title: Journal of Combinatorial Designs – reference: 3.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserlautern. – reference: Bayer, D., 1982. The division algorithm and the Hilbert scheme. Ph. D. Thesis. Harvard University – year: 1997 ident: b6 article-title: Ideals, Varieties and Algorithms – volume: 55 start-page: 19 year: 1944 end-page: 54 ident: b4 article-title: Some results in the theory of quasigroups publication-title: Transactions of the American Mathematical Society – year: 1998 ident: b11 publication-title: Discrete Mathematics Using Latin Squares – volume: vol. 3 year: 1994 ident: b1 publication-title: An Introduction to Gröbner Bases – volume: vol. 4194 start-page: 155 year: 2006 end-page: 165 ident: b9 article-title: Sudokus and Gröbner bases not only a divertimento publication-title: CASC 2006 – volume: 54 start-page: 507 year: 1943 end-page: 519 ident: b2 article-title: Quasigroups I publication-title: Transactions of the American Mathematical Society – reference: Greuel, G.-M., Pfister, G., Schönemann, H., 2005. – ident: 10.1016/j.jsc.2007.07.004_b3 – ident: 10.1016/j.jsc.2007.07.004_b8 – ident: 10.1016/j.jsc.2007.07.004_b7 – volume: vol. 4194 start-page: 155 year: 2006 ident: 10.1016/j.jsc.2007.07.004_b9 article-title: Sudokus and Gröbner bases not only a divertimento – ident: 10.1016/j.jsc.2007.07.004_b10 – year: 1998 ident: 10.1016/j.jsc.2007.07.004_b11 – ident: 10.1016/j.jsc.2007.07.004_b12 – volume: vol. 3 year: 1994 ident: 10.1016/j.jsc.2007.07.004_b1 – ident: 10.1016/j.jsc.2007.07.004_b5 doi: 10.1016/j.jsc.2005.09.007 – year: 1997 ident: 10.1016/j.jsc.2007.07.004_b6 – volume: 54 start-page: 507 year: 1943 ident: 10.1016/j.jsc.2007.07.004_b2 article-title: Quasigroups I publication-title: Transactions of the American Mathematical Society doi: 10.1090/S0002-9947-1943-0009962-7 – volume: 55 start-page: 19 year: 1944 ident: 10.1016/j.jsc.2007.07.004_b4 article-title: Some results in the theory of quasigroups publication-title: Transactions of the American Mathematical Society doi: 10.1090/S0002-9947-1944-0009963-X – volume: 15 start-page: 98 year: 2007 ident: 10.1016/j.jsc.2007.07.004_b13 article-title: Small Latin squares, quasigroups and loops publication-title: Journal of Combinatorial Designs doi: 10.1002/jcd.20105 |
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| Title | Gröbner bases and the number of Latin squares related to autotopisms of order ≤7 |
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