PreprintarXiv licencearXiv
Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters
Abstract
For positive integers m,n, let Am,n=({1,…,m}×Zn)∪{0} have flat addition and multiplication truncated at degree m. We prove that Am,n is finitely based exactly when m≤ 2 or (m,n)=(3,1). Explicit finite bases are supplied throughout this region. Outside it, high-girth hypergraphs with a constant-sum rigidity property yield finite countermodels to every bounded-variable fragment of the equational theory. The proof places no divisibility or coprimality restriction on the parameters.
Related
- Two-element factors and finite bases for additively idempotent semirings
- The finite basis problem for 2× 2 triangular Boolean matrix semirings and incidence semirings
- The finite basis problem for varieties generated by all semirings of a fixed order
- The Finite Basis Problem for Multiplicatively Idempotent Semirings
- The finite basis problem for additively idempotent semirings of order four, IV
BibTeX
@misc{ma2026finite,
title = {{Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters}},
author = {Qirui Ma and Zidong Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26451},
eprint = {2609.26451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26451},
}
RIS
TY - UNPB
TI - Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters
AU - Qirui Ma
AU - Zidong Gao
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26451
UR - https://arxiv.org/abs/2609.26451
AB - For positive integers m,n, let A_m,n=({1,…,m}×Z_n)∪{0} have flat addition and multiplication truncated at degree m. We prove that A_m,n is finitely based exactly when m≤ 2 or (m,n)=(3,1). Explicit finite bases are supplied throughout this region. Outside it, high-girth hypergraphs with a constant-sum rigidity property yield finite countermodels to every bounded-variable fragment of the equational theory. The proof places no divisibility or coprimality restriction on the parameters.
ER -