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PreprintCC BY 4.0arXiv

The unique limit subvariety of the max-plus variety and finite basis properties

  • Xiaolei Shao
  • Mengya Yue
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Abstract

We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring constructed by Shao, Ren and Gao. More generally, we characterize hereditary finite basedness in a finitely defined variety of commutative additively idempotent semirings containing the max-plus variety. Its hereditarily finitely based subvarieties form a finite lattice and admit equational bases involving at most ten variables. We also prove that every proper subvariety of the max-plus variety is locally finite. A finite basis theorem for finite semirings with a multiplicative identity establishes finite basedness of all finite truncations. Finally, we determine the finite basis properties of two families of subvarieties defined by power identities and of subvarieties relatively defined by arbitrary families of identities making individual additive subterms greatest elements.

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Cite

BibTeXRIS
BibTeX
@misc{shao2026unique,
  title = {{The unique limit subvariety of the max-plus variety and finite basis properties}},
  author = {Xiaolei Shao and Mengya Yue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25843},
  eprint = {2609.25843},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25843},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The unique limit subvariety of the max-plus variety and finite basis properties
AU  - Xiaolei Shao
AU  - Mengya Yue
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25843
UR  - https://arxiv.org/abs/2609.25843
AB  - We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring constructed by Shao, Ren and Gao. More generally, we characterize hereditary finite basedness in a finitely defined variety of commutative additively idempotent semirings containing the max-plus variety. Its hereditarily finitely based subvarieties form a finite lattice and admit equational bases involving at most ten variables. We also prove that every proper subvariety of the max-plus variety is locally finite. A finite basis theorem for finite semirings with a multiplicative identity establishes finite basedness of all finite truncations. Finally, we determine the finite basis properties of two families of subvarieties defined by power identities and of subvarieties relatively defined by arbitrary families of identities making individual additive subterms greatest elements.
ER  -