Stable finiteness of twisted group rings and noisy linear cellular automata
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arXiv
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| Formato: | Preprint |
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2022
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| _version_ | 1866916487320567808 |
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| author | Phung, Xuan Kien |
| author_facet | Phung, Xuan Kien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06002 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stable finiteness of twisted group rings and noisy linear cellular automata Phung, Xuan Kien Dynamical Systems Distributed, Parallel, and Cluster Computing Rings and Algebras Cellular Automata and Lattice Gases 16S34, 37B10, 37B15, 43A07, 68Q80 F.1.1 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. |
| title | Stable finiteness of twisted group rings and noisy linear cellular automata |
| topic | Dynamical Systems Distributed, Parallel, and Cluster Computing Rings and Algebras Cellular Automata and Lattice Gases 16S34, 37B10, 37B15, 43A07, 68Q80 F.1.1 |
| url | https://arxiv.org/abs/2209.06002 |