Q-graded Hopf quasigroups

Abstract

Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $\pi -$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the construction method and fundamental theorem of them. Thirdly, we research the smash products of $Q$-graded Hopf quasigroups.

$Q$-graded Hopf quasigroups; $Q$-graded Hopf quasimodules; Smash products