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Zappa–Szép products of Garside monoids

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Abstract

A monoid K is the internal Zappa–Szép product of two submonoids, if every element of K admits a unique factorisation as the product of one element of each of the submonoids in a given order. This definition yields actions of the submonoids on each other, which we show to be structure preserving. We prove that K is a Garside monoid if and only if both of the submonoids are Garside monoids. In this case, these factors are parabolic submonoids of K and the Garside structure of K can be described in terms of the Garside structures of the factors. We give explicit isomorphisms between the lattice structures of K and the product of the lattice structures on the factors that respect the Garside normal forms. In particular, we obtain explicit natural bijections between the normal form language of K and the product of the normal form languages of its factors.

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Authors and Affiliations

  1. Centre for Research in Mathematics, Western Sydney University, Locked Bag 1797, Penrith, NSW, 2751, Australia

    Volker Gebhardt & Stephen Tawn

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  1. Volker Gebhardt
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  2. Stephen Tawn
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Correspondence to Volker Gebhardt.

Additional information

Both authors acknowledge support under Australian Research Council’s Discovery Projects funding scheme (Project Number DP1094072). Volker Gebhardt acknowledges support under the Spanish Projects MTM2010-19355 and MTM2013-44233-P.

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Gebhardt, V., Tawn, S. Zappa–Szép products of Garside monoids. Math. Z. 282, 341–369 (2016). https://doi.org/10.1007/s00209-015-1542-4

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  • DOI: https://doi.org/10.1007/s00209-015-1542-4

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