Weak quasi-entwining structures

Abstract

In this paper  we introduce the notion of weak quasi-entwining structure as a generalization of quasi-entwining structures and weak entwining structures. Also, we formulate the notions of weak cleft extension, weak Galois extension, and weak Galois extension with normal basis associated to a weak quasi-entwining structure.  Moreover, we prove that, under some suitable conditions, there exists an equivalence between  weak Galois extensions with normal basis and weak cleft extensions. As  particular instances, we recover some results previously proved for Hopf quasigroups, weak Hopf quasigroups and weak Hopf algebras.