We consider numeration systems based on a d-tuple $\mathbf{U}=(U_1,\dots ,U_d)$ of sequences of integers and we define $(\mathbf{U},\mathbb{K})$-regular sequences through $\mathbb{K}$-recognizable formal series, where $\mathbb{K}$ is any semiring. We show that, for any d-tuple U of Pisot numeration systems and any semiring $\mathbb{K}$, this definition does not depend on the greediness of the U-representations of integers. The proof is constructive and is based on the fact that the normalization is realizable by a 2d-tape finite automaton. In particular, we use an ad hoc operation mixing a 2d-tape automaton and a $\mathbb{K}$-automaton in order to obtain a new $\mathbb{K}$-automaton.

中文翻译：

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Pisot-规则序列的鲁棒性

我们考虑基于d元组的计数系统$\mathbf{ü}=（{ü}_{\mathrm{1个}}，\dots ，{ü}_d）$ 整数序列，我们定义 $（\mathbf{ü}，\mathbb{ķ}）$-常规序列 $\mathbb{ķ}$-可识别的正式系列，其中 $\mathbb{ķ}$是任何半环。我们证明，对于Pisot计数系统的任何d元组U和任何半环$\mathbb{ķ}$，这个定义不依赖于整数U表示的贪婪性。该证明是有建设性的，并且基于以下事实：归一化可以通过2d磁带有限自动机实现。特别是，我们使用临时操作将2d胶带自动机和$\mathbb{ķ}$-automaton以获取新的 $\mathbb{ķ}$-自动机。