## Dependence Relation In Pregeometries (2000)

Venue: | Algebra Universalis |

Citations: | 2 - 2 self |

### BibTeX

@ARTICLE{Grossberg00dependencerelation,

author = {Rami Grossberg and Olivier Lessmann},

title = {Dependence Relation In Pregeometries},

journal = {Algebra Universalis},

year = {2000},

volume = {44},

pages = {pp}

}

### OpenURL

### Abstract

The aim of this paper is to set a foundation to separate geometric model theory from model theory. Our goal is to explore the possibility to extend results from geometric model theory to non first order logic (e.g. L ! 1 ;! ). We introduce a dependence relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This happens when one is trying to lift forking to nonelementary classes, in contexts where there exists pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2].

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Citation Context ...d guidance and support. The second author is supported by the Swiss National Science Foundation. 1 2 RAMI GROSSBERG AND OLIVIER LESSMANN Finite Equivalence Relation Theorem. Later Baldwin in his book =-=[Bl]-=- presented an axiomatic treatment of forking in stable theories. This allowed Baldwin to derive abstractly Shelah's Pairs Lemma from the other properties of forking. Following these ideas, it has now ... |

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Citation Context ...dence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], =-=[Sh3]-=-, [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different from the first order c... |

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Citation Context ...y classes. A major problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], =-=[HaSh]-=-, [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary clas... |

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Citation Context ...classification theory for nonelementary classes. A major problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], =-=[GrLe1]-=-, [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]... |

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Citation Context ... a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], =-=[MaSh]-=-, [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different from the first ... |

18 |
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Citation Context ...elation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], =-=[Sh47]-=-, [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different from the first order case. In ... |

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Citation Context ... Shelah classification theory for nonelementary classes. A major problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], =-=[GrHa]-=-, [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] a... |

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Citation Context ...X ), is the cardinality of a basis for cl(X ). Fact 5. Using the axioms of pregeometry, one can show that for every set, bases exist and that the dimension is well-defined see for example Appendix in =-=[Gr]-=- Definition 6. Let G = (W ;cl)be a pregeometry. (1) A bijection f:W ! W is an automorphism of G if for every a 2 W and A W we have a 2 cl(A ) if and only if f(a)2 cl(f[A ]): We denote Aut A (G )the se... |

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Citation Context ...e introduction of Chapter III of [Sh b], S. Shelah states what is important about the forking relation is that it satisfies properties (1)--(7). S. Shelah stated another property named by S. Buechler =-=[Bu]-=- the Pairs Lemma (see Proposition 17 for the statement) as one of the basic properties of forking, which was proved in [Sh b] using the Date: April 25, 2000. This is part of the second author's PhD th... |

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Citation Context ...ation theory for nonelementary classes. A major problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], =-=[GrLe2]-=-, [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The sit... |

5 |
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Citation Context ...blem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], =-=[KlSh]-=-, [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different ... |

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4 |
Countable models of ω1-categorical theories in admissible languages
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Citation Context ...or problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], =-=[Ki]-=-, [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very di... |

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3 |
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Citation Context ... a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], =-=[MaSh]-=-, [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different from the first ... |

3 |
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Citation Context ...elation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], =-=[Sh47]-=-, [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different from the first order case. In ... |

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2 |
Strong splitting in stable homogeneous models preprint
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Citation Context ...heories. This happens when one is trying to lift forking to nonelementary classes, in contexts where there exists pregeometries but not necessarily a well-behaved dependence relation (see for example =-=[HySh]-=-). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2]. 1. INTRODUCTIO... |

2 | Geometric Stability Theory Oxford - Pillay - 1996 |

2 |
Foreman and Otmar Spinas, The spectrum of the Γ-invariant of a bilinear space
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Citation Context ... proof. 5. SOME “SET THEORY” In this section, we gather several observations with a set-theoretic flavor. The next theorem is a generalization of a lemma from J. Baumgartner, M. Foreman and O. Spinas =-=[BFS]-=-. Although the proof is easy, it does not follow from the analog theorem involving models as we do not have control over the cardinality of the closures. The value of this theorem is that it makes it ... |

1 |
First order theories of abstract dependence relation
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(Show Context)
Citation Context ...inside the pregeometry, while the original dependence relation inducing the pregeometry is not as well behaved (see [HySh]). A similar endeavor was attempted by John Baldwin in the early eighties. In =-=[Bl1]-=-, J.Baldwin examined some pregeometries and several dependence relations in the first order case. From a pregeometry, he defines the relation a ^ B C , by a 2 cl(B [ C ) cl(B ). He did not however int... |

1 |
Foreman and Otmar Spinas, The spectrum of the -invariant of a bilinear space
- Baumgartner, Matt
- 1997
(Show Context)
Citation Context ... proof. 5. SOME "SET THEORY" In this section, we gather several observations with a set-theoretic flavor. The next theorem is a generalization of a lemma from J. Baumgartner, M. Foreman and =-=O. Spinas [BFS]-=-. Although the proof is easy, it does not follow from the analog theorem involving models as we do not have control over the cardinality of the closures. The value of this theorem is that it makes it ... |

1 |
Categoricity over P for first order T or categoricity for 2 L ! 1 ! can stop at @ k while holding for @ 0 ;;@ k 1
- Hart, Shelah
- 1990
(Show Context)
Citation Context ...y classes. A major problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], =-=[HaSh]-=-, [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary clas... |

1 |
Countable models of ! 1 -categorical theories in admissible languages
- Kierstead
(Show Context)
Citation Context ...or problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], =-=[Ki]-=-, [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very di... |

1 |
and Saharon Shelah. Categoricity for T
- Kolman
(Show Context)
Citation Context ...blem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], =-=[KlSh]-=-, [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different ... |

1 |
Forking in pregeometries, part II: Group configuration
- Lessmann
(Show Context)
Citation Context ...ee for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in =-=[Le2]-=-. 1. INTRODUCTION The notion of forking is at the center of stability theory. Forking is a dependence relation discovered by S. Shelah. It satisfies the following properties in the first order stable ... |