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We give foundational results for the model theory of the ring of finiteadeles over a number field, construed as a restricted product of local fields.In contrast to Weispfenning we work in the language of ring theory, and varioussortings interpretable therein. In particular we give a systematic treatment ofthe product valuation and the valuation monoid. Deeper results are given forthe adelic version of Krasner's hyperfields, relating them to theBasarab-Kuhlmann formalism.

Some supplements to Feferman-Vaught related to the model theory of adeles