Cleft and Galois extensions associated with a weak Hopf quasigroup

In this paper we introduce the notions of cleft and Galois (with normal basis) extension associated with a weak Hopf quasigroup. We show that, under suitable conditions, both notions are equivalent. As a particular instance we recover the classical results for (weak) Hopf algebras. Moreover, taking into account that weak Hopf quasigroups generalize the notion of Hopf quasigroup, we obtain the definitions of cleft and Galois (with normal basis) extension associated with a Hopf quasigroup and we get the equivalence between these extensions in this setting.