Morphisms of naturally valenced association schemes and quotient schemes

作者：Xu Bangteng^*

来源：Journal of Algebra, 2010, 324(9): 2382-2404.

DOI：10.1016/j.jalgebra.2010.08.008

• Association schemes
• Scheme rings
• Morphisms
• Scheme ring homomorphisms
• Closed subsets
• Normal closed subsets
• Quotient subsets
• Quotient schemes

Let S and (S) over tilde be naturally valenced association schemes on sets X and (X) over tilde, respectively, and let phi be a (combinatorial) morphism from (X. S) to ((X) over tilde. (S) over tilde). In Xu (2009) [X2], a necessary and sufficient condition was given for phi to induce an algebra homomorphism from the scheme ring CS to the scheme ring C (S) over tilde. The present paper provides new techniques with which this result can be proved without assuming ker(phi) 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.

• 出版日期2010-11-1