Finite bases and joins for semirings defined by the divisibility order
Abstract
We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of one letter is finitely based, whereas one associated with a linear word is finitely based exactly when the word has length at most two. A hypergraph preservation lemma yields the nonfinite basis result and extends it to intervals of varieties. We establish a sharp containment criterion between the power and linear families, and determine the finite basis property of every join of two varieties generated by one member of each family. The unrestricted power family generates the nonfinitely based max-plus variety. In contrast, the unrestricted linear family has a finite basis, as does its join with each finite power member. We also realize a previously known six-element limit semiring as a quotient of a subsemiring of the eight-element linear-word semiring. This gives a proper nonfinitely based subvariety and resolves the corresponding minimality question.
Related
- Two-element factors and finite bases for additively idempotent semirings
- Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters
- Finite bases for Brandt semirings and naturally semilattice-ordered inverse semigroups
- Finite bases for full power semirings of finite nilpotent semigroups
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BibTeX
@misc{shao2026finite,
title = {{Finite bases and joins for semirings defined by the divisibility order}},
author = {Xiaolei Shao and Zidong Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26396},
eprint = {2609.26396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26396},
}
RIS
TY - UNPB
TI - Finite bases and joins for semirings defined by the divisibility order
AU - Xiaolei Shao
AU - Zidong Gao
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26396
UR - https://arxiv.org/abs/2609.26396
AB - We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of one letter is finitely based, whereas one associated with a linear word is finitely based exactly when the word has length at most two. A hypergraph preservation lemma yields the nonfinite basis result and extends it to intervals of varieties. We establish a sharp containment criterion between the power and linear families, and determine the finite basis property of every join of two varieties generated by one member of each family. The unrestricted power family generates the nonfinitely based max-plus variety. In contrast, the unrestricted linear family has a finite basis, as does its join with each finite power member. We also realize a previously known six-element limit semiring as a quotient of a subsemiring of the eight-element linear-word semiring. This gives a proper nonfinitely based subvariety and resolves the corresponding minimality question.
ER -