Morphisms of naturally valenced association schemes and quotient schemes (Q618253)

Zieschang introduced the notion of morphisms for association schemes. Morphisms of association schemes may not induce algebra homomorphisms between the scheme rings of association schemes. In this paper it is proved: Theorem 1.8. Let \(S\) and \(\tilde S\) be naturally valenced association schemes on sets \(X\) and \(\tilde X\) and let \(\phi: (X,S)\to (\tilde X,\tilde S)\) be a combinatorial morphism. Then \(\phi\) induces a scheme ring homomorphism \(\varphi: \mathbb{C}S \to \mathbb{C}\tilde S\) if and only if the following two conditions are satisfied. {\parindent=6mm \begin{itemize}\item[(i)]\(\ker(\phi)\) is a normal closed subset of \(S\). \item[(ii)]For any \(s\in S\), \(n_{\phi(s)}=n_{s^T}\), where \(T=\ker(\phi)\). \end{itemize}}