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Functoriality of Bose-Mesner algebras and profinite association schemes

We show that taking the set of primitive idempotents of commutative association schemes is a functor from the category of commutative association schemes with surjective morphisms to the category of finite sets with surjective partial functions. We then consider projective systems of commutative association schemes consisting of surjections (which we call profinite association schemes), for which Bose-Mesner algebra is defined, and describe a Delsarte theory on such schemes. This is another method for generalizing association schemes to those on infinite sets, related with the approach by Barg and Skriganov. Relation with $(t,m,s)$-nets and $(t,s)$-sequences is studied. We reprove some of the results of Martin-Stinson from this viewpoint.