Quadratic normalisation in monoids

Patrick Dehornoy 1 Yves Guiraud 2, 3
2 PI.R2 - Design, study and implementation of languages for proofs and programs
IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : In the general context of presentations of monoids, we study normalisation processes that are determined by their restriction to length-two words. Garside's greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter called the class measuring the complexity of the normalisation of length-three words, we analyse the normalisation of longer words and describe a number of possible behaviours. We fully axiomatise normalisations of class (4, 3), show the convergence of the associated rewriting systems, and characterise those deriving from a Garside family.
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Patrick Dehornoy, Yves Guiraud. Quadratic normalisation in monoids. International Journal of Algebra and Computation, World Scientific Publishing, 2016, 26 (5), pp.935-972. ⟨10.1142/S0218196716500399⟩. ⟨hal-01141226v2⟩

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