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

We give foundational results for the model theory of the ring of finite
adeles 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 various
sortings interpretable therein. In particular we give a systematic treatment of
the product valuation and the valuation monoid. Deeper results are given for
the adelic version of Krasner's hyperfields, relating them to the
Basarab-Kuhlmann formalism.