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PreprintCC BY 4.0arXiv

Forking and invariant types in monadic NIP theories

  • Michael C. Laskowski
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Abstract

We obtain a strong decomposition theorem for invariant global types in a monadically NIP theory. From this, we prove that if an n-type tp(a̅/MC) does not fork over M then a̅=(f̅,d̅) where tp(d̅/MC) is M-definable and tp(f̅/MCd̅) is finitely satisfied in M. Over arbitrary base sets B, we prove that an n-type tp(a̅/BC) does not fork over B if and only if tp(ai/BC) does not fork over B for each singleton ai∈a̅. We show that monadically NIP theories satisfy density of definable types among non-forking extensions.

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Cite

BibTeXRIS
BibTeX
@misc{laskowski2026forking,
  title = {{Forking and invariant types in monadic NIP theories}},
  author = {Michael C. Laskowski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25129},
  eprint = {2609.25129},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25129},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Forking and invariant types in monadic NIP theories
AU  - Michael C. Laskowski
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.25129
UR  - https://arxiv.org/abs/2609.25129
AB  - We obtain a strong decomposition theorem for invariant global types in a monadically NIP theory. From this, we prove that if an n-type tp(a̅/MC) does not fork over M then a̅=(f̅,d̅) where tp(d̅/MC) is M-definable and tp(f̅/MCd̅) is finitely satisfied in M. Over arbitrary base sets B, we prove that an n-type tp(a̅/BC) does not fork over B if and only if tp(a_i/BC) does not fork over B for each singleton a_i∈a̅. We show that monadically NIP theories satisfy density of definable types among non-forking extensions.
ER  -