A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety
Abstract
We present an explicit infinite equational basis for the six-element additively idempotent semiring TR6 and prove that TR6 is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by TR6, showing that it forms a four-element chain. Our results demonstrate that the variety generated by TR6 is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, V(TR6) is the first explicit limit subvariety of the variety generated by the max-plus algebra N. Moreover, TR6 is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring SR6, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by SR6 and TR6, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.
Related
- The six-element additively idempotent semiring B21 is inherently nonfinitely based
- Inherently nonfinitely based additively idempotent semirings
- Nonfinitely based intervals of power semiring varieties
- The variety generated by all additively idempotent semirings of order four
- Two-element factors and finite bases for additively idempotent semirings
BibTeX
@misc{shao2026nonfinitely,
title = {{A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety}},
author = {Xiaolei Shao and Miaomiao Ren and Zidong Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09220},
eprint = {2609.09220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09220},
}
RIS
TY - UNPB
TI - A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety
AU - Xiaolei Shao
AU - Miaomiao Ren
AU - Zidong Gao
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.09220
UR - https://arxiv.org/abs/2609.09220
AB - We present an explicit infinite equational basis for the six-element additively idempotent semiring TR_6 and prove that TR_6 is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by TR_6, showing that it forms a four-element chain. Our results demonstrate that the variety generated by TR_6 is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, V(TR_6) is the first explicit limit subvariety of the variety generated by the max-plus algebra N. Moreover, TR_6 is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring SR_6, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by SR_6 and TR_6, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.
ER -