Suprema and infima of association schemes (Q1598810)

Let \(F\) be a partition of a finite set \(\Omega\). In the vector space \(\mathbb{R}^\Omega\) of all real-valued functions on \(\Omega\), denote by \(F_i\) the characteristic function of the \(i\)th class of \(F\). Then the vectors \(F_i\) form a basis for the subspace \(V_F\) consisting of all real functions which are constant on each class of \(F\). Let \(F\) and \(G\) are two partitions of \(\Omega\). We say that \(F\) is finer than \(G\), or that \(F\) is nested in \(G\) if every class of \(F\) is contained in a class of \(G\). This is written \(F\leq G\) and \(\leq\) is a partial order. Classes of supremum \(F\vee G\) are the connected components of the graph on \(\Omega\) which has an edge between vertices \(\alpha\) and \(\beta\) if either \(\alpha\) and \(\beta\) are contained in the same class of \(F\) or \(\alpha\) and \(\beta\) are contained in the same class of \(G\). Classes of infimum \(F\wedge G\) are the non-empty intersections of classes of \(F\) with classes of \(G\). We have \(V_{F\vee G}=V_F\cap V_G\) and \(V_F+V_G\leq V_{F\wedge G}\). If \(A\) and \(B\) are association schemes on a finite set \(\Theta\), then their supremum \(A\vee B\) is also an association scheme on \(\Theta\) (Theorem 2). Let \(\Theta\) be a finite set and let \(F\) be a partition of \(\Theta\times \Theta\). If there is any association scheme \(A\) on \(\Theta\) with \(A\leq F\) then there is a unique coarsest association scheme \(B\) on \(\Theta\) with \(B\leq F\) (Theorem 3). Infima of association schemes are not well behaved in general. The main result of this paper is Theorem 12, which translates the problem of finding infima from the class association schemes on a finite abelian group \(\Theta\) to the class of blueprints.