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A Homotopical Completion Procedure with Applications to Coherence of Monoids

Authors Yves Guiraud, Philippe Malbos, Samuel Mimram



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Yves Guiraud
Philippe Malbos
Samuel Mimram

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Yves Guiraud, Philippe Malbos, and Samuel Mimram. A Homotopical Completion Procedure with Applications to Coherence of Monoids. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 223-238, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)
https://doi.org/10.4230/LIPIcs.RTA.2013.223

Abstract

One of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the low-dimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting systems that keeps track of the cells generated by confluence diagrams. We extend the Knuth-Bendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids.
Keywords
  • higher-dimensional rewriting
  • presentation of monoid
  • Knuth-Bendix completion
  • Tietze transformation
  • low-dimensional homotopy for monoids
  • coherence

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