Finite bases for Brandt semirings and naturally semilattice-ordered inverse semigroups
Abstract
We prove that Brandt semirings over nontrivial groups with at least two indices admit no identity basis with a bounded number of variables. A uniform family of finite witnesses has size linear in the variable bound and applies to groups of arbitrary cardinality. A commutative construction yields the corresponding result for flat extensions of groups of unbounded exponent and gives an exact finite basis criterion for arbitrary flat group extensions. Structural representations lead to a classification of finite natural inverse semirings and a single identity characterizing locally Clifford natural inverse structure among regular semirings. We also describe products, orthogonal sums, minimum generators, colored graph identities and linear variable complexity. Further results concern infinite generating algebras and inherent nonfinite basability.
Related
- Finite bases for full power semirings of finite nilpotent semigroups
- Canonical resolved-transport decomposition and reconstruction of local-unit-aligned ordered semigroups
- Kobayashi's problem for power semigroups of inverse semigroups
- Normality of marginal inverse subsemigroups in inverse semigroups
- Cartan inverse semigroups and twisted Steinberg algebras
BibTeX
@misc{shao2026finite,
title = {{Finite bases for Brandt semirings and naturally semilattice-ordered inverse semigroups}},
author = {Xiaolei Shao and Mengya Yue},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12014},
eprint = {2610.12014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12014},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite bases for Brandt semirings and naturally semilattice-ordered inverse semigroups
AU - Xiaolei Shao
AU - Mengya Yue
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12014
UR - https://arxiv.org/abs/2610.12014
AB - We prove that Brandt semirings over nontrivial groups with at least two indices admit no identity basis with a bounded number of variables. A uniform family of finite witnesses has size linear in the variable bound and applies to groups of arbitrary cardinality. A commutative construction yields the corresponding result for flat extensions of groups of unbounded exponent and gives an exact finite basis criterion for arbitrary flat group extensions. Structural representations lead to a classification of finite natural inverse semirings and a single identity characterizing locally Clifford natural inverse structure among regular semirings. We also describe products, orthogonal sums, minimum generators, colored graph identities and linear variable complexity. Further results concern infinite generating algebras and inherent nonfinite basability.
ER -