Notre Dame Journal of Formal Logic

Rank and Dimension in Difference-Differential Fields

Ronald F. Bustamante Medina

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Abstract

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model of DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent.

Article information

Source
Notre Dame J. Formal Logic Volume 52, Number 4 (2011), 403-414.

Dates
First available in Project Euclid: 4 November 2011

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1320427645

Digital Object Identifier
doi:10.1215/00294527-1499363

Zentralblatt MATH identifier
05991007

Mathematical Reviews number (MathSciNet)
MR2855879

Subjects
Primary: 11U09: Model theory [See also 03Cxx] 12H05: Differential algebra [See also 13Nxx] 12H10: Difference algebra [See also 39Axx]

Keywords
model theory of fields supersimple theories difference-differential fields definable sets

Citation

Bustamante Medina, Ronald F. Rank and Dimension in Difference-Differential Fields. Notre Dame J. Formal Logic 52 (2011), no. 4, 403--414. doi:10.1215/00294527-1499363. http://projecteuclid.org/euclid.ndjfl/1320427645.


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