Equational theories of min-plus semirings

@misc{zhou2026equational,
  title = {{Equational theories of min-plus semirings}},
  author = {Yilin Zhou and Zidong Gao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03715},
  eprint = {2611.03715},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03715},
}

TY  - UNPB
TI  - Equational theories of min-plus semirings
AU  - Yilin Zhou
AU  - Zidong Gao
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.03715
UR  - https://arxiv.org/abs/2611.03715
AB  - We study the equational theory and subvarieties of the variety 𝒱 generated by (ℕ_0,min,+). We prove that a subvariety of 𝒱 is proper if and only if it is locally finite, equivalently, if and only if it satisfies a power-stabilization identity. Every finite member generates a hereditarily finitely based variety with a finite subvariety lattice. In contrast, the periodic subvarieties defined by x^m=x^m+1 are nonfinitely based for m≥2. We construct a strictly descending chain of nonfinitely based subvarieties and three varieties generated by explicit reduced products, each with a decidable equational theory. The smallest is a limit variety: it is nonfinitely based, and every proper subvariety has an absolute finite identity basis. The subvariety lattice of 𝒱 has cardinality continuum. In fact, the interval between the first reduced-product variety and the subvariety defined by x^2=x^3 already has this cardinality, and all its members are nonfinitely based. We also give a hereditarily finitely based example with an infinite subvariety lattice. The constructions use finite convex models and recognition quotients.
ER  -