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Heirs of box types in polynomially bounded structures. (English) Zbl 1187.03035

The author deals with heirs of types in o-minimal structures. His general aim is to find invariants of the types witnessing heritage, or algebraic constructions producing heirs explicitly. This analysis is developed over real closed fields and, more generally, over polynomially bounded o-minimal structures \(M\). In this particular setting, the author already provided a complete description of heirs in terms of invariance groups and invariance rings of 1-types (for \(p\) a 1-type, the invariance group of \(p\) if the convex group \(G(p) : = \{a \in M : a + p = p \}\), while the invariance ring of \(p\) is the convex valuation ring \(\{ b \in M : b \cdot G(p) \subseteq G(p) \}\)).
Here box types are considered (still over a polynomially bounded \(M\)). A box type of \(M\) is an \(n\)-type \(p\) of dimension \(n\) of \(M\) which is uniquely determined by its projections \(p_1, \dots, p_n\) to the coordinate axes (then \(p\) can also be denoted as \((p_1, \dots, p_n)\)). Various examples of box types are proposed, and two methods to build box types are presented. For instance, it is shown that \((p_1,\dots, p_n)\) is a box type when the \(p_i\) have distinct invariance rings.
A notion of heir for box types is introduced in a suitable way. The main theorem characterizes the heirs \((q_1, \dots, q_n)\) of a box type \((p_1, \dots, p_n)\) in terms of the location of the invariance groups and invariance rings of the \(q_i\) compared with the location of those of the \(p_i\). This leads to various structure theorems for subsets of \(M^k\) definable in certain expansions \({\mathcal M}\) of \(M\) by convex subsets of the line. Moreover it is shown that, after naming constants, this structure \({\mathcal M}\) is model-complete provided \(M\) is model-complete.

MSC:

03C64 Model theory of ordered structures; o-minimality
13J30 Real algebra
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