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A Continuum in the Lattice of Semiring Varieties: The Interval [V(S), V(S0)]

  • Zidong Gao
  • Miaomiao Ren
  • Xianzhong Zhao
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Abstract

For an additively idempotent semiring (ai-semiring) S, let S0 denote the ai-semiring obtained from S by adjoining a new element $0$. In this paper, we develop an approach to investigate the interval [V(S), V(S0)] of ai-semiring varieties between the variety generated by S and that generated by S0. We establish a general sufficient condition under which this interval has the cardinality of the continuum. This is applied in particular to [V(S7), V(S70)], where S7 is a $3$-element ai-semiring and is a nonfinitely based algebra of the smallest possible order, thereby resolving an open problem proposed by Jackson, Ren, and Zhao (J. Algebra \textbf{611} (2022), 211--245). The same conclusion holds for [V(B21), V((B21)0)], where B21 is the ai-semiring whose multiplicative reduct is the $6$-element Brandt semigroup. We also present a sufficient condition for the nonfinite basis property in ai-semiring varieties. As a corollary, we obtain a new proof of Dolinka's theorem (Internat. J. Algebra Comput. \textbf{17} (2007), no.~8, 1537--1551) that the $7$-element ai-semiring (B21)0 has no finite basis for its identities.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026continuum,
  title = {{A Continuum in the Lattice of Semiring Varieties: The Interval $[\mathsfV(S), \mathsfV(S^0)]$}},
  author = {Zidong Gao and Miaomiao Ren and Xianzhong Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24514},
  eprint = {2609.24514},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24514},
}
RIS
TY  - UNPB
TI  - A Continuum in the Lattice of Semiring Varieties: The Interval [V(S), V(S^0)]
AU  - Zidong Gao
AU  - Miaomiao Ren
AU  - Xianzhong Zhao
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24514
UR  - https://arxiv.org/abs/2609.24514
AB  - For an additively idempotent semiring (ai-semiring) S, let S^0 denote the ai-semiring obtained from S by adjoining a new element $0$. In this paper, we develop an approach to investigate the interval [V(S), V(S^0)] of ai-semiring varieties between the variety generated by S and that generated by S^0. We establish a general sufficient condition under which this interval has the cardinality of the continuum. This is applied in particular to [V(S_7), V(S_7^0)], where S_7 is a $3$-element ai-semiring and is a nonfinitely based algebra of the smallest possible order, thereby resolving an open problem proposed by Jackson, Ren, and Zhao (J. Algebra \textbf{611} (2022), 211--245). The same conclusion holds for [V(B_2^1), V((B_2^1)^0)], where B_2^1 is the ai-semiring whose multiplicative reduct is the $6$-element Brandt semigroup. We also present a sufficient condition for the nonfinite basis property in ai-semiring varieties. As a corollary, we obtain a new proof of Dolinka's theorem (Internat. J. Algebra Comput. \textbf{17} (2007), no.~8, 1537--1551) that the $7$-element ai-semiring (B_2^1)^0 has no finite basis for its identities.
ER  -