Year: 2026
Type: preprint
Abstract: The variety of semirings satisfying $x^2\approx x$ is hereditarily finitely based in the signature consisting of addition and multiplication, without designated constants. Multiplication may be noncommutative, and the semirings need not be finite. A subdirect decomposition reduces the proof to the subvarieties defined by $3x\approx x$ and $3x\approx2x$. In the latter case, we express each polynomial identity using a common doubled part and isolated word occurrences. Four fixed identities align the first- and last-occurrence words of these occurrences. Induction on their content extracts protected identities. The associated word identities define a band variety, to which the finite basis theorem for bands applies.
Source: arXiv (Cornell University)
Authors: Wang, Aifa, Wang, Lili
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G6489001854
OpenAlex ID: W7223085786