Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Forking and invariant types in regular ordered Abelian groups

Akash Hossain

TL;DR

The paper provides a complete characterization of forking in regular ordered Abelian groups by reducing the ROAG case to DOAG via Presburger elimination and a detailed, valuation-based block analysis. The core method introduces ad-hoc valuations to decompose tuples into Archimedean and ramified blocks, constructs global invariant extensions for blocks, and then glues them into a global invariant type; it shows that for DOAG, non-forking coincides with cut-based (valuation) independence, and extends this to ROAG with precise invariant-extension criteria. A precise unary criterion emerges: forking of tp(C/AB) over A is controlled by the behavior of each C-definable singleton, yielding a dimension-one perspective on forking in this setting. The paper also develops a robust framework of invariance, boundedness, and the space of invariant/global extensions, together with examples that illuminate the relationships between forking, sequential independence, and Presburger-type dependence. The results lay groundwork for a systematic analysis of forking in broader classes of ordered abelian groups, with clear connections to Dolich-independence and Presburger logic.

Abstract

We give a characterization of forking in regular ordered Abelian groups. In particular, we prove that the type of C over AB does not fork over A if and only if the type over AB of each C-definable singleton does not fork over A in these structures.

Forking and invariant types in regular ordered Abelian groups

TL;DR

The paper provides a complete characterization of forking in regular ordered Abelian groups by reducing the ROAG case to DOAG via Presburger elimination and a detailed, valuation-based block analysis. The core method introduces ad-hoc valuations to decompose tuples into Archimedean and ramified blocks, constructs global invariant extensions for blocks, and then glues them into a global invariant type; it shows that for DOAG, non-forking coincides with cut-based (valuation) independence, and extends this to ROAG with precise invariant-extension criteria. A precise unary criterion emerges: forking of tp(C/AB) over A is controlled by the behavior of each C-definable singleton, yielding a dimension-one perspective on forking in this setting. The paper also develops a robust framework of invariance, boundedness, and the space of invariant/global extensions, together with examples that illuminate the relationships between forking, sequential independence, and Presburger-type dependence. The results lay groundwork for a systematic analysis of forking in broader classes of ordered abelian groups, with clear connections to Dolich-independence and Presburger logic.

Abstract

We give a characterization of forking in regular ordered Abelian groups. In particular, we prove that the type of C over AB does not fork over A if and only if the type over AB of each C-definable singleton does not fork over A in these structures.
Paper Structure (30 sections, 98 theorems, 77 equations)

This paper contains 30 sections, 98 theorems, 77 equations.

Table of Contents

  1. Forking in DOAG
  2. Cut-independence
  3. Notation
  4. Valuations
  5. Outline of the proof
  6. Relation to Dolich-independence
  7. How to glue types via ad-hoc valuations
  8. Basic definitions and classification of the cuts
  9. Weak orthogonality
  10. Tensor product
  11. How to build a block extension
  12. The Archimedean case
  13. Archimedean groups
  14. The ramified case
  15. Gluing everything together
  16. ...and 15 more sections

Key Result

Theorem 1

Let $M\models$ROAG, $A$, $B$ parameter subsets of $M$, and $c=c_1...c_n\in M^n$. Let $\kappa=\max(|AB|, 2^{\aleph_0})^+$, and suppose ${M}$ is $\kappa$-saturated and strongly-$\kappa$-homogeneous. If $M$ is discrete, interpret $1$ as its least positive element, else interpret $1$ as $0$ (this is the Moreover, $\textup{tp}(c/AB)$ admits an $\textup{Aut}(M/A)$-invariant extension inside $S(M)$ if an

Theorems & Definitions (257)

  • Theorem 1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.5
  • Example 1.6
  • Lemma 1.7
  • Corollary 1.8
  • Proposition 1.9
  • proof
  • Definition 1.10
  • ...and 247 more