Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters
Q Ma, Z Gao - arXiv preprint arXiv:2609.26451, 2026 - arxiv.org
Q Ma, Z Gao
arXiv preprint arXiv:2609.26451, 2026•arxiv.orgFor positive integers\(m, n\), let\[A_ {m, n}=(\{1,\ldots, m\}\times\Z_n)\cup\{0\}\] have flat
addition and multiplication truncated at degree\(m\). We prove that\(A_ {m, n}\) is finitely
based exactly when\(m\le 2\) 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.
addition and multiplication truncated at degree\(m\). We prove that\(A_ {m, n}\) is finitely
based exactly when\(m\le 2\) 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.
For positive integers , let \[ A_{m,n}=(\{1,\ldots,m\}\times\Z_n)\cup\{0\} \] have flat addition and multiplication truncated at degree . We prove that is finitely based exactly when or . 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.
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