Weak Hopf quasigroups and matched pairs of quasigroupoids

In this paper we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that if $({\sf A}, {\sf H})$ is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid ${\sf A}\bowtie {\sf H}$ called the double cross product of ${\sf A}$ and ${\sf H}$. Also, we show that, if a quasigroupoid ${\sf B}$ admits an exact factorization, there exists a matched pair of quasigroupoids $({\sf A}, {\sf H})$ and an isomorphism of quasigroupoids between ${\sf A}\bowtie {\sf H}$ and ${\sf B}$. Finally, if ${\mathbb K}$ is a field, we show that every matched pair of quasigroupoids $({\sf A}, {\sf H})$ induce, thanks to the quasigroupoid magma construction, a pair $({\mathbb K}[{\sf A}], {\mathbb K}[{\sf H}])$ of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup ${\mathbb K}[{\sf A}]\bowtie{\mathbb K}[{\sf H}]$ isomorphic to ${\mathbb K}[{\sf A}\bowtie {\sf H}]$ as weak Hopf quasigroups.