A computational approach to analyze the Hadamard quasigroup product
Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-03-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023164?viewType=HTML |
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| author | Raúl M. Falcón Víctor Álvarez José Andrés Armario María Dolores Frau Félix Gudiel María Belén Güemes |
| author_facet | Raúl M. Falcón Víctor Álvarez José Andrés Armario María Dolores Frau Félix Gudiel María Belén Güemes |
| author_sort | Raúl M. Falcón |
| collection | DOAJ |
| description | Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular. |
| first_indexed | 2024-04-09T13:47:21Z |
| format | Article |
| id | doaj.art-b5ecfe3f8c624aa6b241e2f41f42d4e4 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-09T13:47:21Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-b5ecfe3f8c624aa6b241e2f41f42d4e42023-05-09T01:28:35ZengAIMS PressElectronic Research Archive2688-15942023-03-013163245326310.3934/era.2023164A computational approach to analyze the Hadamard quasigroup productRaúl M. Falcón 0Víctor Álvarez1José Andrés Armario2María Dolores Frau3Félix Gudiel 4María Belén Güemes5Department of Applied Mathematics I, Universidad de Sevilla, SpainDepartment of Applied Mathematics I, Universidad de Sevilla, SpainDepartment of Applied Mathematics I, Universidad de Sevilla, SpainDepartment of Applied Mathematics I, Universidad de Sevilla, SpainDepartment of Applied Mathematics I, Universidad de Sevilla, SpainDepartment of Applied Mathematics I, Universidad de Sevilla, SpainBased on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular.https://www.aimspress.com/article/doi/10.3934/era.2023164?viewType=HTMLhadamard productquasigrouplatin squarelatin transversalisomorphismcomputer algebra system |
| spellingShingle | Raúl M. Falcón Víctor Álvarez José Andrés Armario María Dolores Frau Félix Gudiel María Belén Güemes A computational approach to analyze the Hadamard quasigroup product Electronic Research Archive hadamard product quasigroup latin square latin transversal isomorphism computer algebra system |
| title | A computational approach to analyze the Hadamard quasigroup product |
| title_full | A computational approach to analyze the Hadamard quasigroup product |
| title_fullStr | A computational approach to analyze the Hadamard quasigroup product |
| title_full_unstemmed | A computational approach to analyze the Hadamard quasigroup product |
| title_short | A computational approach to analyze the Hadamard quasigroup product |
| title_sort | computational approach to analyze the hadamard quasigroup product |
| topic | hadamard product quasigroup latin square latin transversal isomorphism computer algebra system |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023164?viewType=HTML |
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