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Definable sets up to definable bijections in Presburger groups. (English) Zbl 1404.03032

In this paper, the authors generalize some of the results of the first author [J. Symb. Log. 68, No. 1, 153–162 (2003; Zbl 1046.03019)] about the classification of definable sets up to definable bijections in the ordered group of the integers \(\mathbb Z\). The “Presburger groups” of the title are \(\mathbb Z\)-groups, i.e. elementary extensions of \(\mathbb Z\) in the first-order language of abelian ordered groups. The motivation of the authors stems from motivic integration, where \(\mathbb Z\)-groups would appear as valuation groups of valued fields, in particular the work of E. Hrushovski and D. Kazhdan [Geom. Funct. Anal. 17, No. 6, 1924–1967 (2008; Zbl 1213.03046)] where the valuation groups are divisible.
The main results are formulated in terms of Grothendieck semirings. In \(\mathbb Z\), two definable sets are in definable bijections iff they have same cardinality and same dimension. The general situation is too involved to state precisely here. The main ingredients are:
1)
that every definable set is in definable bijection with a finite disjoint union of products of intervals;
2)
new invariants generalizing cardinality and dimension: “hyper-cardinality”, and a family of relative dimensions using the work of A. Fornasiero [Ann. Pure Appl. Logic 162, No. 7, 514–543 (2011; Zbl 1233.03037)].
Roughly, in hyper-cardinality, cardinality is replaced by certain polynomial functions in a definable way, and the relative dimensions are based on a combination of the (model-theoretic) algebraic closure operator and the convex closure (for subgroups) operator. For bounded definable sets, hyper-cardinality is a full invariant up to definable bijection. The authors also obtain results on definable families of bounded sets.

MSC:

03C60 Model-theoretic algebra
03C10 Quantifier elimination, model completeness, and related topics
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
13D15 Grothendieck groups, \(K\)-theory and commutative rings
16Y60 Semirings
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References:

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