The L-R-Smash Products For Hopf Quasigroups

This paper mainly studies the L-R-smash product for Hopf quasigroups, and the L-R-smash coproduct for Hopf coquasigroups by duality.This paper is consisted of the following three chapters:In chapter1, we first introduce the development of Hopf algebras and Hopf quasigroups. And the main results studied in this paper will be introduced.In chapter2, we give the definition of biquasimodule Hopf quasigroup. A necessary and sufficient condition for making an L-R-smash product into a Hopf quasigroup is given.In chapter3, we give the definition of biquasicomodule Hopf coquasigroup and then get a necessary and sufficient condition for making an L-R-smash coproduct into a Hopf coquasigroup.