论文标题
扭曲组环和嘈杂线性细胞自动机的稳定有限
Stable finiteness of twisted group rings and noisy linear cellular automata
论文作者
论文摘要
对于线性的非均匀细胞自动机（NUCA），它们是群体$ g $的线性CA的局部扰动，并且在任意字段$ k $上是有限维矢量空间$ v $ v $，我们研究了他们的Dedekind有限属性属性，也被称为直接有限属性，即直接有限属性，即左或右或右图的无限性。我们说，$ g $是$ l^1 $ - surjunctive，resp。有限$ l^1 $ - 外部，如果所有此类线性核在稳定的注入性时都会自动汇总，请自动汇总。另外，$ k $是有限的。同时，我们将环$ d^1（k [g]）$介绍为笛卡尔产品$ k [g] \ times（k [g]）[g] $作为添加剂组，但乘法在第二个组件中扭曲。环$ d^1（k [g]）$自然包含group und $ k [g] $，我们在每个字段$ k $的稳定有限$ l^1 $ l^1 $ surjunctivity the Group $ g $方面的稳定性表征，例如，当$ G $是$ G $的残留有限的或最初的次数下属或最初的子量后，它持有。我们的结果扩展了已知的结果。
For linear non-uniform cellular automata (NUCA) which are local perturbations of linear CA over a group universe $G$ and a finite-dimensional vector space alphabet $V$ over an arbitrary field $k$, we investigate their Dedekind finiteness property, also known as the direct finiteness property, i.e., left or right invertibility implies invertibility. We say that the group $G$ is $L^1$-surjunctive, resp. finitely $L^1$-surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in addition $k$ is finite. In parallel, we introduce the ring $D^1(k[G])$ which is the Cartesian product $k[G] \times (k[G])[G]$ as an additive group but the multiplication is twisted in the second component. The ring $D^1(k[G])$ contains naturally the group ring $k[G]$ and we obtain a dynamical characterization of its stable finiteness for every field $k$ in terms of the finite $L^1$-surjunctivity of the group $G$, which holds for example when $G$ is residually finite or initially subamenable. Our results extend known results in the case of CA.
