$(G,\phi)$-crossed product on $(G,\phi)$-quasiassociative algebras

The notions of $(G,\phi)$-crossed product and quasicrossed system are introduced in the setting of $(G,\phi)$-quasiassociative algebras, i.e., algebras endowed with a grading by a group $G$, satisfying a “quasiassociative” law. It is presented two equivalence relations, one for quasicrossed systems and another for $(G,\phi)$-crossed products. Also the notion of graded-bimodule in order to study simple $(G,\phi)$-crossed products is studied.