##  [Lascar Rank](/lascar-rank-0) 

 Definition

A stability-theoretic ordinal rank on types that refines other ordinal ranks by incorporating Lascar strong-type distinctions; it is sensitive to the finer equivalence relation given by Lascar strong types and their extension behavior.

 

 

 

 

 

 





## Principle

Principle

Lascar rank is defined using refinements of type-equivalence (Lascar strong types) and chains of extensions that respect these strong-type classes; ordinals are assigned by an inductive scheme similar to other ranks but using Lascar-strong splitting or extension conditions.

 

 

 

 

 





## Demonstration

Demonstration

In contexts where Lascar strong types matter (for example, when fine control of automorphism orbits is needed), Lascar rank can distinguish types that share Morley or U-rank; two types with the same Morley rank may have different Lascar rank because they split into different Lascar strong-type components.

 

 

 

 

## Misapplication

Misapplication

Treating Lascar rank as identical to U-rank or Morley rank without checking strong-type distinctions, or attempting to compute it in settings where Lascar strong types are poorly understood or where the Lascar equivalence relation collapses.

 

 

 

 

 





## Consequence

Consequence

Lascar rank refines classification by detecting subtler differences between types that other ordinal ranks miss; it can expose additional internal symmetry or asymmetry in definable sets and feed into finer structural theorems about automorphism groups and canonical bases.

 

 

 

 

## Reversal

Reversal

Where Lascar rank collapses to coarser ranks, strong-type distinctions are invisible and many fine structural conclusions fail; conversely, when Lascar rank is strictly finer, one gains greater resolution at the cost of added technical complexity.

 

 

 

 

 





## Boundary

Boundary

Defined for types in theories where Lascar strong types and their extension behavior are meaningful; in some theories Lascar strong types coincide with ordinary strong types and the Lascar rank offers no extra information.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between Lascar rank and coarser ranks (U- or Morley-rank): Lascar rank trades simplicity for sensitivity, sometimes distinguishing types that appear identical under other ordinal measures; it is also related to Lascar distance and can be conflated with it if care is not taken.

 

 

 

 

 





## Synthesis

Synthesis

Lascar rank is a refinement of ordinal model-theoretic ranks that incorporates Lascar strong-type information to provide a finer measurement of type complexity and extension behavior, useful when automorphism-sensitive distinctions matter.