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 [Formula: see text] is a matched pair of quasigroupoids, then it is possible to construct a new quasigroupoid [Formula: see text] called the double cross product of [Formula: see text] and [Formula: see text]. Also, we show that, if a quasigroupoid [Formula: see text] admits an exact factorization, then there exists a matched pair of quasigroupoids [Formula: see text] and an isomorphism of quasigroupoids between [Formula: see text] and [Formula: see text]. Finally, if [Formula: see text] is a field, then we show that every matched pair of quasigroupoids [Formula: see text] induces, thanks to the quasigroupoid magma construction, a pair [Formula: see text] of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup [Formula: see text] isomorphic to [Formula: see text] as weak Hopf quasigroups.