Stable domination and weight. (English) Zbl 1252.03091
Concepts of weight and domination are introduced for types in an arbitrary first-order theory (via forking) and studied for types which are dominated by a stable type. If \(p=\mathrm{tp}(\bar a/A)\) is dominated by a stable type over \(A\) it is proved that:
1) \(p\) is domination-equivalent to a stable type (of an infinite tuple) over \(A\); as a witnessing tuple the stable part of \(\bar a\) over \(A\) can be taken;
2) If \(T\) is strong in the sense of [H. Adler, “Strong theories, burden, and weight”, Preprint (2007)] then \(p\) is domination-equivalent to a finite product of stable, weight-1 types;
3) If \(T\) is dependent and \(A\) is a model then \(p\) is stably dominated.
The definition of stable domination here slightly differs from the original one from [D. Haskell, E. Hrushovski and H. D. Macpherson, Stable domination and independence in algebraically closed valued fields. Lecture Notes in Logic 30. Cambridge: Cambridge University Press. (2008; Zbl 1149.03027)] and in the Appendix it is sketched how a proof of the theorem of “descent” (Theorem 4.9 in the book) can be adapted to the present context.
A careful reader may notice that Observation 4.4 is incorrect but it does not affect the other results.
1) \(p\) is domination-equivalent to a stable type (of an infinite tuple) over \(A\); as a witnessing tuple the stable part of \(\bar a\) over \(A\) can be taken;
2) If \(T\) is strong in the sense of [H. Adler, “Strong theories, burden, and weight”, Preprint (2007)] then \(p\) is domination-equivalent to a finite product of stable, weight-1 types;
3) If \(T\) is dependent and \(A\) is a model then \(p\) is stably dominated.
The definition of stable domination here slightly differs from the original one from [D. Haskell, E. Hrushovski and H. D. Macpherson, Stable domination and independence in algebraically closed valued fields. Lecture Notes in Logic 30. Cambridge: Cambridge University Press. (2008; Zbl 1149.03027)] and in the Appendix it is sketched how a proof of the theorem of “descent” (Theorem 4.9 in the book) can be adapted to the present context.
A careful reader may notice that Observation 4.4 is incorrect but it does not affect the other results.
Reviewer: Predrag Tanović (Beograd)
MSC:
| 03C45 | Classification theory, stability, and related concepts in model theory |
Citations:
Zbl 1149.03027References:
| [1] | H. Adler, Strong theories and weight, preprint.; H. Adler, Strong theories and weight, preprint. |
| [2] | Adler, H., A geometric introduction to forking and thorn-forking, J. Math. Log., 9, 1-20 (2009) · Zbl 1211.03051 |
| [3] | D. Haskell, E. Hrushovski, D. Macpherson, Stable domination and independence in algebraically closed valued fields 30 (2008) xii+182.; D. Haskell, E. Hrushovski, D. Macpherson, Stable domination and independence in algebraically closed valued fields 30 (2008) xii+182. · Zbl 1149.03027 |
| [4] | Hasson, A.; Onshuus, A., Stable types in rosy theories, J. Symbolic Logic, 75, 4, 1211-1230 (2010) · Zbl 1247.03053 |
| [5] | E. Hrushovski, A. Pillay, Nip and invariant measures (submitted for publication).; E. Hrushovski, A. Pillay, Nip and invariant measures (submitted for publication). · Zbl 1220.03016 |
| [6] | Hyttinen, T., Remarks on structure theorems for \(\omega_1\)-saturated models, Notre Dame J. Formal Logic, 36, 269-278 (1995) · Zbl 0854.03031 |
| [7] | Kim, B., Forking in simple unstable theories, J. London Math. Soc. (2), 57, 257-267 (1998) · Zbl 0922.03048 |
| [8] | B. Kim, Simple first order theories, Notre Dame, 2001.; B. Kim, Simple first order theories, Notre Dame, 2001. |
| [9] | Lascar, D.; Poizat, B., An introduction to forking, J. Symbolic Logic, 44, 330-350 (1979) · Zbl 0424.03013 |
| [10] | A. Onshuus, A. Usvyatsov, On dp-minimality, strong stability and weight, J. Symbolic Logic (in press).; A. Onshuus, A. Usvyatsov, On dp-minimality, strong stability and weight, J. Symbolic Logic (in press). · Zbl 1252.03091 |
| [11] | A. Onshuus, A. Usvyatsov, Thorn orthogonality and domination in unstable theories (submitted for publication).; A. Onshuus, A. Usvyatsov, Thorn orthogonality and domination in unstable theories (submitted for publication). · Zbl 1277.03032 |
| [12] | Pillay, A., (Geometric Stability Theory. Geometric Stability Theory, Oxford Logic Guides, vol. 32 (1996), The Clarendon Press Oxford University Press: The Clarendon Press Oxford University Press New York), Oxford Science Publications · Zbl 0871.03023 |
| [13] | S. Shelah, Strongly dependent theories, 2007 (submitted for publication) (Sh863).; S. Shelah, Strongly dependent theories, 2007 (submitted for publication) (Sh863). · Zbl 1371.03043 |
| [14] | Shelah, S., (Classification Theory and the Number of Nonisomorphic Models. Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, vol. 92 (1990), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam) · Zbl 0713.03013 |
| [15] | Shelah, S., Dependent first order theories, continued, Israel J. Math., 173, 1-60 (2009) · Zbl 1195.03040 |
| [16] | Usvyatsov, A., Generically stable types in dependent theories, J. Symbolic Logic, 74, 216-250 (2009) · Zbl 1181.03040 |
| [17] | Wagner, F. O., (Simple Theories. Simple Theories, Mathematics and its Applications, vol. 503 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) · Zbl 0948.03032 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
