Fundamentals of quasigroup Hopf group-coalgebras.

A large class of algebras (possibly nonassociative) with group-coalgebraic structures, called quasigroup Hopf group-coalgebras, is introduced and studied. Quasigroup Hopf group-coalgebras provide a unifying framework for the classical Hopf algebras and Hopf group-coalgebras as well as Hopf quasigroups. Then, basic results similar to those in Hopf algebras H are proved, such as anti-(co) multiplicativity of the antipode S: H → H, and S² = id if H is commutative or cocommutative. [ABSTRACT FROM AUTHOR]