Rewriting systems and biautomatic structures for Chinese, hypoplactic, and Sylvester monoids

This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and Sylvester monoids. For Chinese monoids, we first give new presentations via finite complete rewriting systems, using more lucid constructions and proofs than those given independently by Chen & Qui and Güzel Karpuz; we then construct biautomatic structures. For hypoplactic monoids, we construct finite complete rewriting systems and biautomatic structures. For Sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures. Consequently, the monoid algebras corresponding to monoids of these classes are automaton algebras in the sense of Ufnarovskij. Acknowledgements: During the research that led to the this paper, the first author was supported by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT (Fundação para a Ciência e a Tecnologia) under the project PEst-C/MAT/UI0144/2011 and through an FCT Ciência 2008 fellowship. For the second and third authors, this work was developed within the project POCTI-ISFL1-143 of CAUL, supported by FCT and FEDER.