Suprema and infima of association schemes

Partitions on a set are partially ordered by nesting: partition F is nested in partition G if every class of F is contained in a class of G. In this partial order, the supremum of two partitions is the finest partition which nests them both. An association scheme on a finite set \Omega is just a partition of \Omega \times \Omega which satisfies some technical conditions. An incomplete-block design is said to be partially balanced with respect to a given association scheme if that scheme is nested in the concurrence partition of the design. The supremum of two association schemes is also an association scheme. As a consequence, if an incomplete-block design is partially balanced at all then there is a unique coarsest association scheme with respect to which it is partially balanced. Infima of association schemes are not well behaved in general. However, some special classes of association scheme of interest to statisticians permit interesting conclusions about infima.