Distributive laws and Hopf quasigroups

In this paper we introduce the notion of $a$-monoidal distributive law between two Hopf quasigroups $A$ and $H$. We prove that every $a$-monoidal distributive law induce a product on $A\otimes H$, called the wreath product, thanks to which $A\otimes H$ becomes in a Hopf quasigroup. Finally, using this construction, we show that double cross products of Hopf quasigroups, cross products of Hopf quasigroups with a skew pairing between them, Hopf quasigroups defined by the twisted double method, smash products of Hopf quasigroups and twisted smash products of Hopf quasigroups are examples of wreath products associated to $a$-monoidal distributive laws.