Model theory of finite-by-Presburger Abelian groups and finite extensions of pp-adic fields

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are finite-by-Presburger and interpret the higher residue rings of the local field. We apply these results to give a new proof of the model completeness in the ring language of a local field of characteristic zero (a result that follows also from work of Prestel-Roquette)

Model Completeness for Henselian Fields with finite ramification valued in a ZZ-Group

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. We apply this to prove that every infinite algebraic extension of the field of $p$-adic numbers $\Bbb Q_p$ with finite ramification is model-complete in the language of rings. For this, we give a necessary and sufficient condition for model-completeness of the theory of a perfect pseudo-algebraically closed field with pro-cyclic absolute Galois group

Enrichments of Boolean Algebras: a uniform treatment of some classical and some novel examples

We give a unified treatment of the model theory of various enrichments of infinite atomic Boolean algebras, with special attention to quantifier-eliminations, complete axiomatizations and decidability. A classical example is the enrichment by a predicate for the ideal of finite sets, and a novel one involves predicates giving congruence conditions on the cardinality of finite sets. We focus on three examples, and classify them by expressive power

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]] inside F_p((t)), which works uniformly for all $p$ and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Z_p inside Q_p uniformly for all p. For any fixed finite extension of Q_p, we give an existential formula and a universal formula in the ring language which define the valuation ring

Commutative unital rings elementarily equivalent to prescribed product rings

The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component structures. Here, by closely related methods, but in the special setting of commutative unital rings, we obtain a kind of converse allowing us to determine in interesting cases, when a commutative unital R is elementarily equivalent to a nontrivial product of a family of commutative unital rings R_i. We use this in the model theoretic analysis of residue rings of models of Peano Arithmetic