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PreprintCC BY 4.0arXiv

Finite BM-algebras and fixed-point combinators

  • Takuma Imamura
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Abstract

We study algebras for the BM-, BI-, and BMI-fragments of combinatory logic, where B xyz β†’w x(yz), M x β†’w xx and I x β†’w x. We prove that every finite BM-algebra π’œ contains an element Y satisfying Yx = x(Yx) for all x∈ A. Moreover, a single closed BM-term represents such an element in every finite BM-algebra. On the other hand, we show that there exists an infinite BMI-algebra with no such element. Consequently, the variety of BM-algebras is not generated by finite BM-algebras. We also characterise finite BI-algebras as finite semigroups with two distinguished left units B and I, and finite BMI-algebras as finite idempotent semigroups with three distinguished left units B, M and I.

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Cite

BibTeXRIS
BibTeX
@misc{imamura2026finite,
  title = {{Finite $\mathbfBM$-algebras and fixed-point combinators}},
  author = {Takuma Imamura},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37108},
  eprint = {2610.37108},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37108},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite BM-algebras and fixed-point combinators
AU  - Takuma Imamura
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37108
UR  - https://arxiv.org/abs/2610.37108
AB  - We study algebras for the BM-, BI-, and BMI-fragments of combinatory logic, where B xyz β†’_w x(yz), M x β†’_w xx and I x β†’_w x. We prove that every finite BM-algebra π’œ contains an element Y satisfying Yx = x(Yx) for all x∈ A. Moreover, a single closed BM-term represents such an element in every finite BM-algebra. On the other hand, we show that there exists an infinite BMI-algebra with no such element. Consequently, the variety of BM-algebras is not generated by finite BM-algebras. We also characterise finite BI-algebras as finite semigroups with two distinguished left units B and I, and finite BMI-algebras as finite idempotent semigroups with three distinguished left units B, M and I.
ER  -