论文标题
由$ x^n \大约x $定义的Burnside AI隔离品种
The Burnside ai-semiring variety defined by $x^n\approx x$
论文作者
论文摘要
令$ {\ bf sr}（n，1）$表示由身份$ x^n \大约x $定义的ai隔离变体，其中$ n> 1 $。我们表征了$ {\ bf sr}（n，1）$的半圣经子变量的所有细分不可约成员。基于此结果，我们证明$ {\ bf sr}（n，1）$是遗传性有限的（分别是遗传性，遗传性生成），并且仅当$ n <4 $，并且$ {\ bf sr}（\ bf sr}（n，1）$的subvarieties lattice lattice lattice lattice of -if and if if and if and Bun -if and if uf and Born -if $ n $ n $ n <4 $。另外，我们表明$ {\ bf sr}的所有本地有限成员的类（n，1）$形成了一个品种，因此我们肯定回答了$ {\ bf sr}（n，1）$的受限伯恩赛德问题。此外，我们还提供了Gajdoš和Ku红车获得的主要结果的简化证明（Semigroup论坛80：92---104，2010）。
Let ${\bf Sr}(n, 1)$ denote the ai-semiring variety defined by the identity $x^n\approx x$, where $n>1$. We characterize all subdirectly irreducible members of a semisimple subvariety of ${\bf Sr}(n, 1)$. Based on this result, we prove that ${\bf Sr}(n, 1)$ is hereditarily finitely based (resp., hereditarily finitely generated) if and only if $n<4$ and that the lattice of subvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, we show that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms a variety and so we affirmatively answer the restricted Burnside problem for ${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main result obtained by Gajdoš and Kuřil (Semigroup Forum 80: 92--104, 2010).
