Mathematische Zeitschrift

, Volume 282, Issue 1–2, pp 341–369 | Cite as

Zappa–Szép products of Garside monoids



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.

Mathematics Subject Classification

Primary 20F36 Secondary 20M13 06F05 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Centre for Research in MathematicsWestern Sydney UniversityPenrithAustralia

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