A hypergroup join is a
hypergroup formed by the union of a discrete hypergroup with a compact
hypergroup. The compact hypergroup is a subhypergroup of the join, but the
convolution on the discrete hypergroup is changed in the join. A characterization of
compact joins in terms of their dual objects is given which leads to a simpler criteria
for their existence. In particular, it is shown that if a compact abelian join has a dual
which is a hypergroup, then the dual is also a join. Examples of joins are provided
from the study of conjugacy classes of certain semi-direct products of compact groups
and a method is described for constructing non-dualizable compact abelian
hypergroups.
