[Paper Review] Orthogonality and domination in unstable theories

Alf Onshuus, Alexander Usvyatsov|arXiv (Cornell University)|Sep 21, 2009

Advanced Topology and Set Theory3 references3 citations

TL;DR

This paper investigates orthogonality, domination, and weight in unstable theories, particularly focusing on rosy and dependent theories. It establishes that in strongly dependent rosy theories, every type has finite thorn-weight, and shows that dividing in dependent theories can be witnessed by Morley sequences, providing a structural understanding of forking and type decomposition in unstable contexts.

ABSTRACT

In the first part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. Then we try to develop analogous theory for arbitrary dependent theories.

Motivation & Objective

• To extend concepts like domination, regularity, and weight from stable and simple theories to rosy and super-rosy theories.
• To investigate whether strong dependence implies finite thorn-weight in rosy theories, addressing a key question in model theory.
• To develop a natural notion of forking weight in arbitrary dependent theories and clarify its connection to strong dependence.
• To establish that dividing in dependent theories is always witnessed by Morley sequences, a structural result with broad implications.
• To provide a strong decomposition theorem for types of finite rank in rosy theories, enabling analysis via minimal and regular types.

Proposed method

• Adapts classical proofs of stability-theoretic results to the context of þ-forking and thorn-independence in rosy theories.
• Uses mutual indiscernibility and Morley sequences as central tools to analyze forking and weight in dependent theories.
• Applies the notion of strong dividing and 2-cc-strong dividing to characterize when formulas divide over a base set.
• Employs the concept of rudimentary finite weight and connects it to strong dependence via the second author's earlier work.
• Develops technical tools such as Proposition 3.6 to analyze type decomposition in finite-rank rosy theories.
• Proves that in dependent theories, dividing over an extension base is witnessed by a Morley sequence, generalizing known results from stable and simple theories.

Experimental results

Research questions

• RQ1 Does every type in a strongly dependent rosy theory have finite thorn-weight?
• RQ2 Can a natural notion of forking weight be defined in arbitrary dependent theories, and how does it relate to strong dependence?
• RQ3 Is dividing in a dependent theory always witnessed by a Morley sequence?
• RQ4 What is the role of mutually indiscernible sequences in understanding forking and weight in dependent theories?
• RQ5 How can types of finite rank in rosy theories be decomposed using regular and minimal types?

Key findings

• In a strongly dependent rosy theory, every type has finite thorn-weight, confirming a conjecture linking strong dependence and finiteness of weight.
• Dividing in a dependent theory over an extension base is always witnessed by a Morley sequence, a result that strengthens understanding of forking behavior.
• The paper establishes a strong decomposition theorem for types of finite rank in rosy theories, showing that such types can be analyzed via regular and minimal types.
• The existence of mutually indiscernible sequences in dependent theories is characterized, providing a foundation for further analysis of forking and weight.
• The connection between rudimentary finite weight and finite weight is clarified, showing that rudimentary finite weight implies finite weight in rosy theories.
• The results provide a complementary framework to prior work on minimal types in super-rosy theories, particularly in the context of theories interpretable in o-minimal structures.