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 ( A, H) is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid A H called the double cross product of A and H. Also, we show that, if a quasigroupoid B admits an exact factorization, there exists a matched pair of quasigroupoids ( A, H) and an isomorphism of quasigroupoids between A H and B. Finally, if K is a field, we show that every matched pair of quasigroupoids ( A, H) induce, thanks to the quasigroupoid magma construction, a pair ( K[ A], K[ H]) of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup K[ A] K[ H] isomorphic to K[ A H] as weak Hopf quasigroups.