The class of Abelian p-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory whose models are each bi-interpretable with the disjoint union of an Abelian p-group and a pure set (and so that every Abelian p-group is bi-interpretable with a model of ) using computable infinitary formulas. This answers a question of Knight by giving an example of an elementary first-order theory of “Ulm type”: Any two models, low for , and with the same computable infinitary theory, are isomorphic. It also gives a new example of an elementary first-order theory whose isomorphism problem is -complete but not Borel complete.
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