# Q-graded Hopf quasigroups

## TL;DR

This paper introduces Q-graded Hopf quasigroups, a new algebraic structure generalizing existing concepts, and explores their properties, representations, and smash products, contributing to the field of algebraic systems.

## Contribution

It defines Q-graded Hopf quasigroups, studies their properties, representations, and smash products, expanding the understanding of algebraic generalizations.

## Key findings

- Defined Q-graded Hopf quasigroups and their properties
- Developed the theory of Q-graded Hopf quasimodules and their construction
- Analyzed smash products of 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.