Morphisms of naturally valenced association schemes and quotient schemes

Let S and be naturally valenced association schemes on sets X and , respectively, and let ϕ be a (combinatorial) morphism from (X,S) to . In Xu (2009) [X2], , a necessary and sufficient condition was given for ϕ to induce an algebra homomorphism from the scheme ring CS to the scheme ring . The present paper provides new techniques with which this result can be proved without assuming ker(ϕ) to be finite. To do this, we will first need to prove that for any normal closed subset T of S, whether T is finite or infinite, the quotient S//T is a naturally valenced association scheme on the set X/T. We will also need to discuss scheme ring homomorphisms of naturally valenced association schemes, and prove some isomorphism theorems without assuming the kernels of the scheme ring homomorphisms to be finite. As a direct consequence, for a naturally valenced commutative association scheme S on a set X and any closed subset T of S, the quotient S//T is a naturally valenced commutative association scheme on the set X/T. The approach in this paper is different from Xu (2009) [X2], and quasi-algebraic morphisms of naturally valenced association schemes are also studied.