Association schemes in which the thin residue is a finite cyclic group

In this article, we investigate association schemes S (on finite sets) in which each element s satisfies ss⁎s={s}. It is shown that these schemes are schurian if the partially ordered set of the intersections of the closed subsets s⁎s of S with s∈S is distributive. (A scheme is said to be schurian if it arises (in a well-understood way) from a transitive permutation group.) It is also shown that, if these schemes are schurian, the transitive permutation group from which they arise have subnormal one-point stabilizers. As a consequence of the first result one obtains that schemes are schurian if their thin residue is thin and has a distributive normal closed subset lattice (normal subgroup lattice). This implies, for instance, that schemes are schurian if their thin residue is a cyclic group.