Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Shao, Xiaolei, Yue, Mengya
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221071016