`ev.gebhardt@uws.edu.au`

In this talk, I will report on joint work with Stephen Tawn. \smallskip

The notion of Zappa-Széep products generalises those of direct and semidirect products; the key property is that every element of the Zappa-Szép product can be written uniquely as a product of two elements, one from each factor, in a given order.

In the context of Garside monoids, Zappa-Szép products were studied by Picantin; he used the term *crossed products*.
Given a family $M_1,\ldots,M_\ell$ of Garside monoids and a family of maps $\Theta_{i,j}:M_i\times M_j\to M_j$ that satisfy some compatibility conditions, Picantin constructs a Garside structure on the set $M_1\times\cdots\times M_\ell$, that is, he considers *external* Zappa-Szép products.

Picantin's construction is fairly technical and not easy to work with in practice; this comes mainly from the compatibility conditions for the maps $\Theta_{i,j}$, which are needed to ensure that the way in which the factors $M_i$ are made to interact is consistent and the external Zappa-Szép product is well-defined.

We were primarily interested in *decomposing* a given Garside monoid into simpler components that are also Garside monoids. In this situation, the way in which the potential factors interact is defined by the ambient monoid and there is no need for explicit compatibility conditions. It was therefore natural for us to consider *internal* Zappa-Szép products:

**Definition 1**
Given a monoid $K$ with two submonoids $G$ and $H$, say that $K$ is
the (internal) *Zappa-Szép product* of $G$ and $H$, written
$K=G\zs H$, if for every $k\in K$ there exist unique $g_1, g_2\in
G$ and $h_1, h_2 \in H$ such that $g_1 h_1 = k = h_2 g_2$.

As Picantin showed that an element of the product can be written uniquely as a product of elements of the factors in a given order, every crossed product is also a Zappa-Szép product in the sense of Definition 1.

Definition 1 implies the existence of left- and right-actions of $G$ on $H$ and vice versa. By exploiting these actions, it is possible to obtain many natural results in a relatively straightforward way; I will show some examples in this talk.

In particular, the Zappa-Szép product $K = G \zs H$ of two monoids is a Garside monoid if and only if both of the factors are Garside monoids. In this case, the factors are standard parabolic submonoids of $K$ and the Garside structures of $K$ can be described in terms of the Garside structures of the factors. We obtain an explicit isomorphism between the lattice structures of $K$ and the product of the lattice structures on the factors, that respects the Garside normal forms. The latter yields bijections between the normal form language of $K$ and the product of the normal form languages of its factors.