in this paper using strongly duplexes we introduce a new class of (semi)hypergroups. the associated (semi)hypergroup from a strongly duplex is called duplex (semi)hypergroup. two computer programs written in matlab show that the two groups $z_{2n}$ and $z_{n}times z_{2}$ produce a strongly duplex and its associated hypergroup is a complementary feasible hypergroup.

The article is devoted to the describtion of quasitriangular structures (universal R-matrices) on cocommutative Hopf algebras. It is known that such structures are concentrated on finite dimensional Hopf subalgebras. In particular, quasitriangular structure on group algebra is defined by the pairs of normal inclusions of an finite abelian group and by invariant bimultiplicative form on it. The ...

We generalize the notion of a Rota-Baxter operator on groups and weight 1 Lie algebras define study cocommutative Hopf algebra $H$. If $H=F[G]$ is group $G$ or $H=U(\mathfrak{g})$ universal enveloping $\mathfrak{g}$, then we prove that operators $H$ are in one to correspondence with corresponding algebras.