Topic
Forking extension
About: Forking extension is a research topic. Over the lifetime, 3 publications have been published within this topic receiving 18 citations.
Papers
TL;DR: A generalized notion of rank is defined for stable theories without dense forking chains, and it is applied to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisationof a forking extension.
Abstract: We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.
11 citations
TL;DR: The cardinal i ( T), which is essential in computing the number of types over a set which is independent over some set, is introduced, and the author expects the reader will discover the importance of via these theorems.
Abstract: In [3], Shelah defined the cardinals κ n (T) and , for each theory T and n ω . κ n (T) is the least cardinal κ without a sequence ( p i ) i of complete n -types such that p i is a forking extension of p j for all i j κ . It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T . (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In this paper, we shall give equivalents of which can be easily handled. In §1 we shall state the definitions of κ n (T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property. In §2 we shall give some conditions on κ, n , and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n . We introduce the cardinal i ( T ), which is essential in computing the number of types over a set which is independent over some set, and show that i ( T ) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems. Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.
5 citations
Posted Content•
TL;DR: In this paper, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: the {\em degree of nonminimality} is the minimum number of realisations of the type required to witness a nonalgebraic forking extension.
Abstract: Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The {\em degree of nonminimality} is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the $U$-rank plus $2$. The Borovik-Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of $U$-rank plus $1$ is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.
5 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 1992 | 1 |
| 1985 | 1 |