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PreprintCC BY 4.0arXiv

On Local Finiteness of Modal K4 Algebras

  • Gabriel Agnew
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Abstract

We study local finiteness for modal K4 algebras via the tunability of their dual general frames. In particular, we provide a sufficient condition for modal K4 algebras to be locally finite by identifying a structure which must be present in non-locally finite modal K4 algebras. We then show that this condition becomes both necessary and sufficient for σ-complete modal K4 algebras. Next, we translate this condition into a pair of order-theoretic conditions on transitive Kripke frames, providing a classification of local finiteness on their dual modal algebras. We further show that the logic of any class of well-founded transitive relations with no infinite antichains has the finite model property, and conclude that the logic of the class of well-quasi orderings has the finite model property.

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Cite

BibTeXRIS
BibTeX
@misc{agnew2026local,
  title = {{On Local Finiteness of Modal K4 Algebras}},
  author = {Gabriel Agnew},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.02941},
  eprint = {2606.02941},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.02941},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Local Finiteness of Modal K4 Algebras
AU  - Gabriel Agnew
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2606.02941
UR  - https://arxiv.org/abs/2606.02941
AB  - We study local finiteness for modal K4 algebras via the tunability of their dual general frames. In particular, we provide a sufficient condition for modal K4 algebras to be locally finite by identifying a structure which must be present in non-locally finite modal K4 algebras. We then show that this condition becomes both necessary and sufficient for σ-complete modal K4 algebras. Next, we translate this condition into a pair of order-theoretic conditions on transitive Kripke frames, providing a classification of local finiteness on their dual modal algebras. We further show that the logic of any class of well-founded transitive relations with no infinite antichains has the finite model property, and conclude that the logic of the class of well-quasi orderings has the finite model property.
ER  -