Finite identity bases for flat semirings of linear words
Abstract
For a set W of nonempty words, let S(W) be the flat semiring formed by the nonempty factors of words in W, together with an absorbing zero. We prove that S(W) has a finite identity basis whenever every word in W is linear, with no restriction on the size of W or on the lengths of its words. In the bounded case its variety is generated by the interval semiring Am≅ S(a1⋯ am), where m is the maximum word length and the letters ai are distinct. In the unbounded case its variety is generated by the interval semiring on all finite intervals of the nonnegative integers. We give explicit finite bases in both cases. The proofs encode nonzero polynomial evaluations by endpoint graphs and derive the required graph identifications using finitely many splicing identities. In particular, the eleven-element semiring S(abcd) is finitely based, providing a counterexample to the length-bound conjectures for S(W) proposed by Gao, Ren and Zhao.
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BibTeX
@misc{wang2026finite,
title = {{Finite identity bases for flat semirings of linear words}},
author = {Aifa Wang Lili Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14349},
eprint = {2609.14349},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14349},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite identity bases for flat semirings of linear words
AU - Aifa Wang Lili Wang
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14349
UR - https://arxiv.org/abs/2609.14349
AB - For a set W of nonempty words, let S(W) be the flat semiring formed by the nonempty factors of words in W, together with an absorbing zero. We prove that S(W) has a finite identity basis whenever every word in W is linear, with no restriction on the size of W or on the lengths of its words. In the bounded case its variety is generated by the interval semiring A_m≅ S(a_1⋯ a_m), where m is the maximum word length and the letters a_i are distinct. In the unbounded case its variety is generated by the interval semiring on all finite intervals of the nonnegative integers. We give explicit finite bases in both cases. The proofs encode nonzero polynomial evaluations by endpoint graphs and derive the required graph identifications using finitely many splicing identities. In particular, the eleven-element semiring S(abcd) is finitely based, providing a counterexample to the length-bound conjectures for S(W) proposed by Gao, Ren and Zhao.
ER -