Elsevier

Journal of Algebra

Volume 324, Issue 9, 1 November 2010, Pages 2382-2404
Journal of Algebra

Morphisms of naturally valenced association schemes and quotient schemes

https://doi.org/10.1016/j.jalgebra.2010.08.008Get rights and content
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Abstract

Let S and S˜ be naturally valenced association schemes on sets X and X˜, respectively, and let ϕ be a (combinatorial) morphism from (X,S) to (X˜,S˜). 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 CS˜. 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.

Keywords

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

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