Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T15:51:20Z No. of bitstreams: 2 Dissertação - Laredo Rennan Pereira Santos - 2015.pdf: 970933 bytes, checksum: 6b8836c42db993ababe18805a5857373 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T15:54:12Z (GMT) No. of bitstreams: 2 Dissertação - Laredo Rennan Pereira Santos - 2015.pdf: 970933 bytes, checksum: 6b8836c42db993ababe18805a5857373 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-19T15:54:12Z (GMT). No. of bitstreams: 2 Dissertação - Laredo Rennan Pereira Santos - 2015.pdf: 970933 bytes, checksum: 6b8836c42db993ababe18805a5857373 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we consider monoid presentations MonhX,Ri, with set of generators X and relations R, defining groups and monoids as equivalence classes of words over X, in relation to a congruence generated by R. Taking R as a rewriting system with respect to an linear ordering of X , the set of words over X, we can apply some rewriting strategies in its laws. We use a version of the Knuth-Bendix method in R to find a confluent rewriting system equivalent to original, when such finite system exist. This new set of relations, denoted by RC(X, R), allows that in MonhX,Ri any element be defined by a unique irreducible word with respect to RC(X, R). We exhibit several examples of the execution of the Knuth-Bendix method from the functions of KBMAG package of the GAP system. Lastly, we set up a sufficient condition so that certain monoid presentations for polycyclic groups be confluent. / Neste trabalho, consideramos apresentações monoidais MonhX,Ri, com conjunto de geradores X e de relações R, definindo grupos e monoides como classes de equivalência de palavras sobre X, em relação a uma congruência gerada por R. Tomando R como um sistema de reescrita com respeito à uma ordenação linear de X , o conjunto de palavras sobre X, podemos aplicar algumas estratégias de reescrita em suas leis. Usamos uma versão do método de Knuth-Bendix em R para encontrar um sistema de reescrita confluente que seja equivalente ao original, quando um tal sistema finito existe. Este novo conjunto de relações, denotado por RC(X, R), permite que em MonhX,Ri qualquer elemento seja definido por uma única palavra irredutível com respeito a RC(X, R). Exibimos diversos exemplos da execução do método de Knuth-Bendix a partir das funções do pacote KBMAG do sistema GAP. Por fim, estabelecemos uma condição suficiente para que certas apresentações monoidais para grupos policíclicos sejam confluentes.

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Cette thèse est consacrée à l’étude des monoïdes plaxiques par une nouvelle approche utilisant des méthodes issues de la réécriture. Ces méthodes sont appliquées à des présentations de monoïdes plaxiques décrites en termes de tableaux de Young, de bases cristallines de Kashiwara et de modèle des chemins de Littelmann. On étudie le problème des syzygies pour la présentation de Knuth des monoïdes plaxiques. En utilisant la procédure de complétion homotopique basée sur les procédures de complétion de Squier et de Knuth–Bendix, on construit des présentations cohérentes de monoïdes plaxiques de type A. Une telle présentation cohérente étend la notion de présentation convergente d’un monoïde par une famille génératrice de syzygies, décrivant toutes les relations entre les relations. On explicite une présentation cohérente finie des monoïdes plaxiques de type A avec les générateurs colonnes. Cependant, cette présentation n’est pas minimale dans le sens que plusieurs de ses générateurs sont superflus. En appliquant la procédure de réduction homotopique, on réduit cette présentation en une présentation cohérente finie qui étend la présentation de Knuth, donnantainsi toutes les syzygies des relations de Knuth. D’une manière plus générale, on étudie des présentations de monoïdes plaxiques généralisés du point de vue de la réécriture. On construit des présentations convergentes finies de ces monoïdes en utilisant les chemins de Littelmann. De plus, on étudie ces présentations pour le type C en termes de bases cristallines de Kashiwara. En introduisant les générateurs colonnes admissibles, on construit une présentation convergente finie du monoïde plaxique de type C avec des relations explicites. Cette approche nous permettrait d’étudier le problème des syzygies des présentations de monoïdes plaxiques en tout type / This thesis focuses on the study of plactic monoids by a new approach using methods issued from rewriting theory. These methods are applied on presentations of plactic monoids given in terms of Young tableaux, Kashiwara’s crystal bases and Littelmann path model. We study the syzygy problem for the Knuth presentation of the plactic monoids. Using the homotopical completion procedure that extends Squier’s and Knuth–Bendix’s completions procedure, we construct coherent presentations of plactic monoids of type A. Such a coherent presentation extends the notion of a presentation of a monoid by a family of generating syzygies, taking into account all the relations among the relations. We make explicit a finite coherent presentation of plactic monoids of type A with the column generators. However, this presentation is not minimal in the sense that many of its generators are superfluous. After applying the homotopical reduction procedure on this presentation, we reduce it to a finite coherent one that extends the Knuth presentation, giving then all the syzygies of the Knuth relations. More generally, we deal with presentations of plactic monoids of any type from the rewriting theory perspective. We construct finite convergent presentations for these monoids in a general way using Littelmann paths. Moreover, we study the latter presentations in terms of Kashiwara’s crystal graphs for type C. By introducing the admissible column generators, we obtain a finite convergent presentation of the plactic monoid of type C with explicit relations. This approach should allow us to study the syzygy problem for the presentations of plactic monoids for any type