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

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

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

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

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

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

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

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

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.