Hopf?-quasigroup

In this paper,we define a new class of weakened Hopf Algebras: Hopf?-quasigroups,and study some of them.Here,we let ? be a discrete group with neutral element.Firstly,we introduce the concept about ?-coalgebras;then Hopf?-quasigroups and Hopf?-quasimodules are defined,we prove the basic properties and the Hopf modules basic theorem of Hopf?-quasigroups;Secondly,by defining the ?-H-quasimodules,we get the definition of ?-Hquasimodules algebra and ?-H-quasimodules coalgebras,and then construct the ?-smash products of Hopf?-quasigroups,we prove the necessary and sufficient conditions of ?-smash products Hopf?-quasigroups;then,by defining the ?-H-comodules,we obtain the definition of ?-H-comodules algebras and quasi ?-H-comodules coalgebras,constructing the?-smash coproducts of Hopf?-quasigroups,we prove the necessary and sufficient conditions of ?-smash coproducts Hopf?-quasigroups;Finally,we construct the ?-smash biproducts of Hopf?-quasigroups,and prove the necessary and sufficient conditions of ?-smash biproducts Hopf?-quasigroups.