##
**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.

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.

Reviewer: Carlo Toffalori (Camerino)

### Keywords:

heir; definable type; polynomially bounded o-minimal structure; Dedekind cut; box type; valuation theory
Full Text:
DOI

### References:

[1] | DOI: 10.1016/j.apal.2008.01.009 · Zbl 1146.03027 |

[2] | Cours de théorie des modèles (1985) |

[3] | DOI: 10.1090/S0273-0979-1984-15249-2 · Zbl 0542.03016 |

[4] | Definable types in o-minimal theories 59 pp 185– (1994) |

[5] | Omitting types in o-minimal theories 51 pp 63– (1986) |

[6] | DOI: 10.1090/S0002-9947-00-02633-7 · Zbl 0982.03021 |

[7] | An introduction to forking 44 pp 330– (1979) |

[8] | Stability in model theory (1987) |

[9] | Weakly semialgebraic spaces (1989) · Zbl 0681.14008 |

[10] | Forking and independence in o-minimal theories 69 pp 215– (2004) |

[11] | Paires de structures o-minimales 63 pp 570– (1998) |

[12] | DOI: 10.1112/S0024611500012648 · Zbl 1062.03029 |

[13] | t-convexity and tame extensions 60 pp 74– (1995) |

[14] | Valuation theoretic content of the Marker–Steinhorn Theorem 69 pp 91– (2004) · Zbl 1080.03025 |

[15] | DOI: 10.1007/s00153-006-0022-2 · Zbl 1116.03033 |

[16] | DOI: 10.1016/j.apal.2004.12.003 · Zbl 1089.03032 |

[17] | Model completeness of o-minimal structures expanded by Dedekind cuts 70 pp 29– (2005) · Zbl 1097.03032 |

[18] | Mathematical logic (1967) · Zbl 0149.24309 |

[19] | Classification theory and the number of non-isomorphic models (1978) |

[20] | Topological properties of sets definable in weakly o-minimal structures (2005) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.