  It is well known that finite commutative association schemes in the sense of
the monograph of Bannai and Ito lead to finite commutative hypergroups with
positive dual convolutions and even dual hypergroup structures. In this paper
we present several discrete generalizations of association schemes which also
lead to associated hypergroups. We show that discrete commutative hypergroups
associated with such generalized association schemes admit dual positive
convolutions at least on the support of the Plancherel measure. We hope that
examples for this theory will lead to the existence of new dual positive
product formulas in near future.
