论文标题
有限小组行动的代数和艾伦伯格和舒特伯格的问题
Algebras from finite group actions and a question of Eilenberg and Schützenberger
论文作者
论文摘要
1976年，Eilenberg和M.-P。 Schützenberger提出了以下可疑的问题：如果$ \ m athbf {a} $是一个有限的代数结构，$σ$是$ \ mathbf {a} $中的所有身份的集合，并且存在有限的子集$ f $ f $ f $ f $ f $和fin fin fid a fin fin，则必须是fin f $ f。 $σ$使得$ f'$和$σ$具有完全相同的有限和无限型号？ （也就是说，$ \ mathbf {a} $的身份必须是“有限的”？），众所周知，对他们的问题的任何反例（如果存在的话）必须以一种特别奇怪的方式有限的。在本文中，我们表明，由劳伦斯和威拉德构建的“本质上的基于非绝对的”代数不会以这种特别奇怪的方式有限地基于有限的，因此不要对Eilenberg和Schützenberger的问题提供反示例。作为推论，我们给出了由群体动作构建的第一个已知固有基于非绝对基于的“自动代数”的示例。
In 1976 S. Eilenberg and M.-P. Schützenberger posed the following diabolical question: if $\mathbf{A}$ is a finite algebraic structure, $Σ$ is the set of all identities true in $\mathbf{A}$, and there exists a finite subset $F$ of $Σ$ such that $F$ and $Σ$ have exactly the same finite models, must there also exist a finite subset $F'$ of $Σ$ such that $F'$ and $Σ$ have exactly the same finite and infinite models? (That is, must the identities of $\mathbf{A}$ be "finitely based"?) It is known that any counter-example to their question (if one exists) must fail to be finitely based in a particularly strange way. In this paper we show that the "inherently nonfinitely based" algebras constructed by Lawrence and Willard from group actions do not fail to be finitely based in this particularly strange way, and so do not provide a counter-example to the question of Eilenberg and Schützenberger. As a corollary, we give the first known examples of inherently nonfinitely based "automatic algebras" constructed from group actions.
