On a class of wreath products of hypergroups and association schemes

We characterize finite hypergroups S (in the sense of Frédéric Marty in Huitième Congres des Mathématiciens, pp. 45–59, 1934) satisfying |pq|=1 for any two elements p and q in S with p≠q ^? in terms of wreath products. The result applies to association schemes of finite valency and provides a corresponding characterization in scheme theory. For association schemes S of finite valency satisfying the above condition, we provide a second characterization, a characterization in terms of the subconstituent algebra of S.