PreprintCC BY 4.0arXiv
The Finite Basis Problem for Multiplicatively Idempotent Semirings
Abstract
The variety of semirings satisfying x2≈ 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≈ x and 3x≈2x. 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.
Related
- The finite basis problem for additively idempotent semirings of order four, IV
- Congruence-simple multiplicatively idempotent semirings and multiplicative divisibility
- The finite basis problem for varieties generated by all semirings of a fixed order
- The finite basis problem for 2× 2 triangular Boolean matrix semirings and incidence semirings
- The finite basis problem for flat semirings Sc(W), Mc(W) and M(W)
BibTeX
@misc{wang2026finite,
title = {{The Finite Basis Problem for Multiplicatively Idempotent Semirings}},
author = {Aifa Wang and Lili Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27463},
eprint = {2610.27463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27463},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Finite Basis Problem for Multiplicatively Idempotent Semirings
AU - Aifa Wang
AU - Lili Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27463
UR - https://arxiv.org/abs/2610.27463
AB - The variety of semirings satisfying x^2≈ 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≈ x and 3x≈2x. 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.
ER -