Abstract

Given a module $\mathcal{T}$ over an F-algebra $\mathcal{A},$ and a basis $\mathcal{B}$ for $\mathcal{T},$ the amenability domain of $\mathcal{B}$ is the subalgebra of $\mathcal{A}$ consisting of all elements $r\in \mathcal{A}$ with row finite matrix representation ${[l_r]}_{\mathcal{B}}$ with respect to $\mathcal{B}.$ A basis $\mathcal{B}$ of $\mathcal{T}$ having amenability domain equal to $\mathcal{A}$ is said to be an amenable basis. When an amenable basis for $\mathcal{T}$ exists, $\mathcal{T}$ is said to be an amenable module. A basis having amenability domain F is said to be contrarian. We show that, under some mild additional hypotheses, amenable modules always have contrarian bases. The collection of amenability domains of bases of a module $\mathcal{T}$ is called the amenability profile of the module $\mathcal{T}.$ The profile is a measurement of the diversity of the bases of $\mathcal{T}$ and serves to sort them according to the extent of their amenability. We consider when profiles of amenable modules are minimal (consisting of only F and $\mathcal{A}$) and when they are maximal (consisting of all the subalgebras of $\mathcal{A}$); the former modules are said to lack discernment and the latter to be full rank. We provide an example of a graph magma algebra without discernment. We also show that $F[x]$ does not lack discernment and that graph magma algebras are never full rank. As an alternative to full rank for graph magma algebras, we consider the largest domain of amenability feasible in that context and introduce Maximal Rank graph magma algebras; we show that such algebras indeed exist.

摘要

给定一个模块 $\mathcal{Ť}$在F代数上$\mathcal{一种}，$ 和基础 $\mathcal{乙}$ 对于 $\mathcal{Ť}，$ 的顺应性域 $\mathcal{乙}$ 是...的子代数 $\mathcal{一种}$ 由所有元素组成 $\mathrm{[R}\in \mathcal{一种}$ 行有限矩阵表示 ${[{升}_{\mathrm{[R}}]}_{\mathcal{乙}}$ 关于 $\mathcal{乙}。$ 基础 $\mathcal{乙}$ 的 $\mathcal{Ť}$ 拥有一个等于 $\mathcal{一种}$据说是可以接受的基础。当适合的基础$\mathcal{Ť}$ 存在， $\mathcal{Ť}$据说是一个不错的模块。具有顺应性域F的基础被认为是逆向的。我们表明，在一些轻微的附加假设下，可服从的模块始终具有逆势基础。模块基础的适应性域的集合$\mathcal{Ť}$ 称为模块的顺应性配置文件 $\mathcal{Ť}。$ 轮廓是对基础的多样性的度量 $\mathcal{Ť}$并根据其适用性对它们进行分类。我们考虑何时合适的模块的轮廓最小（仅由F和$\mathcal{一种}$），以及当它们最大时（由的所有子代数组成） $\mathcal{一种}$）; 据说前者模块缺乏洞察力，而后者则是完整的。我们提供了一个图岩浆代数的例子，没有区别。我们还表明$F[X]$不缺乏洞察力，图岩浆代数从来都不是最高级的。作为图岩浆代数的全秩的替代方法，我们认为在这种情况下最大的顺应性域是可行的，并介绍了最大秩图岩浆代数。我们证明了这样的代数确实存在。