轨道有限线性方程组的可解性
Solvability of orbit-finite systems of linear equations
AI总结：
本文研究带原子集合上的轨道有限线性方程组，提出一个适用于任意域或交换环的可解性判定过程，并证明轨道有限向量空间存在轨道有限基。
摘要：
我们在带原子集合的设定下研究轨道有限线性方程组。我们的主要贡献是一个判定此类系统可解性的决策过程。该过程在温和的有效性假设下适用于每个域（甚至交换环），并将给定的轨道有限系统归约为若干个有限系统：一般情况下为指数多个，但当输入系统的原子维数固定时为多项式多个。为获得该过程，我们进一步推进了由轨道有限集合生成的向量空间理论，并证明每个这样的向量空间都允许一个轨道有限基。这一基本性质是我们研究中的关键工具，同时也应具有更广泛的兴趣。
英文摘要：
We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under mild effectiveness assumptions, and reduces a given orbit-finite system to a number of finite ones: exponentially many in general, but polynomially many when atom dimension of input systems is fixed. Towards obtaining the procedure we push further the theory of vector spaces generated by orbit-finite sets, and show that each such vector space admits an orbit-finite basis. This fundamental property is a key tool in our development, but should be also of wider interest.
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