Lascar Rank in Expansions — Draconitic Research

Abstract

We prove that the SU-rank of a supersimple theory is no greater than that of any of its supersimple expansions. A proof for finite rank theories already exists in the literature, but non-continuity presents a problem at limit ranks. We prove the non-decreasing of SU-rank in expansions type-by-type by reducing to finding, in the base theory, certain trees of forking extensions in which the formulas and dividing numbers which witness dividing have a sufficiently uniform pattern. In the case of countable ranks, we can give an especially nice description of this uniformity: for a countable ordinal α, any type of SU-rank at least α in a supersimple theory has a chain of forking extensions in the reverse order type of α.

Voltage-Increment-Based Low-Rank Representation with Polynomial Chaos Expansions for Probabilistic Voltage Assessment Under Multiple Contingencies
Yuzhi Wang, Xingyu Lin, Junzhou Wang +3
Mathematics2 Oct 2026Peer-reviewedCC BY 4.0General mathematicsCrossref
Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
Honghuai Fang, Tian Zhou
arXiv30 Aug 2026PreprintarXiv licenceFull textAlgebra & number theory Geometry & topologyarXiv
Character expansions and affine Jacobi-Trudi identities
Ilse Fischer, Moritz Gangl, Álvaro Gutiérrez +1
arXiv6 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Transcendence of dendric beta-expansions with pure discrete spectrum
Jihyuk Seo, Wolfgang Steiner
arXiv19 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theory Dynamical systemsarXivHAL
On differentiability in weakly o-minimal expansions of fields
Juan Pablo Acosta, Yatir Halevi, Assaf Hasson +2
arXiv2 Oct 2026PreprintCC BY 4.0Full textLogic & foundationsarXiv

Cite

BibTeX RIS

BibTeX

@misc{lange2026lascar,
  title = {{Lascar Rank in Expansions}},
  author = {Michael Lange},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14698},
  eprint = {2609.14698},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14698},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Lascar Rank in Expansions
AU  - Michael Lange
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14698
UR  - https://arxiv.org/abs/2609.14698
AB  - We prove that the SU-rank of a supersimple theory is no greater than that of any of its supersimple expansions. A proof for finite rank theories already exists in the literature, but non-continuity presents a problem at limit ranks. We prove the non-decreasing of SU-rank in expansions type-by-type by reducing to finding, in the base theory, certain trees of forking extensions in which the formulas and dividing numbers which witness dividing have a sufficiently uniform pattern. In the case of countable ranks, we can give an especially nice description of this uniformity: for a countable ordinal α, any type of SU-rank at least α in a supersimple theory has a chain of forking extensions in the reverse order type of α.
ER  -

Abstract

Related

Cite