Lascar rank

Lascar rank
Name	Lascar rank
Occupation	Mathematical concept

Lascar rank is a notion in model theory used to measure complexity of types and definable sets in complete first-order theories. It refines and interacts with other invariants such as Morley rank, U-rank, Shelah rank, and stability-theoretic notions, and plays a role in classification theory developed by Shelah and contributors like Morley, Pillay, and Hrushovski. Originating in work connected with descriptive set theory and geometric stability, the concept has applications to simple theories, stable groups, and analyzability in structures related to algebraic geometry and number theory such as fields with operators and differentially closed fields.

Definition and basic properties

Lascar rank is defined for complete types in a complete first-order theory in terms of ordinal-valued ranks that generalize Cantor–Bendixson or Cantor rank constructions used in descriptive set theory and topology. For a complete type p over a parameter set A one assigns an ordinal (or ∞) by iterating definable forking or dividing patterns similarly to how Morley rank is defined; key properties include monotonicity, invariance under automorphisms fixing parameters, and additivity or subadditivity under independent extensions. The rank interacts with notions introduced by Shelah in classification theory such as superstability, and with concepts studied by Morley, Zilber, Hrushovski, Pillay, and Lascar himself. In well-behaved contexts Lascar rank coincides with Morley rank or U-rank; in others it refines U-rank and gives tighter distinctions among types as in work by Poizat, Wagner, and Buechler.

Historical development and motivation

The origin of Lascar rank traces to efforts in the 1960s–1980s to classify complete theories via ordinal-valued invariants after the foundational contributions of Morley, Shelah, and Lachlan. Developments in stable theory by Morley and Baldwin–Lachlan motivated finer analytic tools; Lascar introduced a rank to distinguish types in unstable but tame contexts, influenced by work of Cohen, Sacks, and Cantor on ranks in set theory. Subsequent advances by Pillay, Poizat, Hrushovski, Wagner, and Buechler integrated Lascar rank into the apparatus of simple theories and geometric stability, paralleling themes found in Grothendieck-style classification in algebraic geometry and model-theoretic treatments of differential algebra pioneered by Kolchin and later by Cassidy and Singer.

Calculation and examples

Computing Lascar rank for specific theories often uses known classifications: in algebraically closed fields (ACF) of characteristic 0 the rank of a complete type corresponding to an irreducible variety equals the Zariski dimension, echoing correspondences studied by Chevalley and Weil. In differentially closed fields (DCF) the rank relates to Kolchin polynomial degree and to work of Cassidy and Hrushovski on differential algebraic varieties. For groups of finite Morley rank studied by Cherlin and Borovik, Lascar rank typically matches Morley rank; in o-minimal structures like expansions of the real field by Wilkie the rank behaves like dimension in semi-algebraic and subanalytic geometry connected to work of van den Dries. Counterexamples demonstrating divergence from Morley rank appear in constructions by Hrushovski and Poizat producing exotic structures where Lascar rank detects forking patterns absent in classical algebraic examples.

Relation to other model-theoretic ranks

Lascar rank is interwoven with Morley rank, U-rank (Lascar–Urank), Shelah's ranks such as Rk and D-rank, and notions of weight and thorn-forking introduced by Adler, Kim, and Pillay. In stable theories Morley rank, Lascar rank, and U-rank frequently coincide; in simple theories U-rank and Lascar rank differ but relate via inequalities and preserving properties under non-forking extensions studied by Shelah and Wagner. Connections to geometric stability notions explored by Zilber, Hrushovski, and Pillay show how ranks reflect internality, analyzability, and modularity of types, with examples from compact complex manifolds, finite Morley rank groups, and difference fields (ACFA) studied by Chatzidakis and Hrushovski.

Applications in stability theory and classification

Lascar rank serves as a tool in proving structural theorems about stable and simple theories, including categoricity results à la Morley and Baldwin–Lachlan, stability spectra analyses by Shelah, and simplicity-theoretic decompositions by Kim and Pillay. It aids the study of definable groups and fields in works by Poizat, Hrushovski, and Wagner, contributes to the analysis of Zariski geometries developed by Zilber, and appears in model-theoretic treatments of arithmetic geometry connected to Faltings, Serre, and Mazur via definable sets in fields. In classification theory it helps to distinguish tame settings where geometric methods apply from wild settings exemplified by Hrushovski amalgamation constructions and model companions like ACFA and DCF.

Variants and generalizations

Variants include ordinal-valued refinements, Lascar–Urank modifications for simple theories, and analogues using thorn-forking and Kim-forking developed by Adler, Kim, and Pillay. Generalizations adapt Lascar-type ranks to abstract elementary classes and ℵ_0-stable contexts, and to metric structures in continuous model theory as in work by Ben Yaacov, Berenstein, and Henson. Further extensions connect to homological and categorical invariants in topos-theoretic and o-minimal frameworks studied by van den Dries, Pillay, and Grothendieck-influenced approaches to definable categories.

Category:Model theory