Supernormal subhypergroup

ANALOGY: This is an analogue in hypergroups of the subgroup property:
View other analogues of normality | View other analogues in hypergroups of subgroup properties

History

Origin

The term supernormal subhypergroup was introduced by Bloom and Heyer in their paper Cpnvergence of convolution products of probability measures on hypergroups.

Definition

A subhypergroup $H$ of a hypergroup $K$ is said to be supernormal if $x*H*\overline{x}=H$ for every element $x\in K$.

Analogy

The subhypergroup property of being supernormal is analogous to the subgroup property of being normal, as per the following definition: a subgroup $H$ of a group$K$ is said to be normal if $xH{x}^{-1}=H$ for any $x\in K$.

Relation with other properties

Weaker properties

• Normal subhypergroup

References

• Convergence of convolutions products of probability measures on hypergroups by W. R. Bloom and H. Heyer, Rend. Mat. Appl. 7, Serial 2, 547-563, 1982
• Connectivity and supernormality results for hypergroups by Richard C. Vrem, Math. Z. 1987