抽象数制中的正则序列和同步序列
Regular sequences and synchronized sequences in abstract numeration systems
摘要：
$b$ -正则序列的概念在2002年由Maes和Rigo推广到抽象数制。它们的定义基于$\mathcal{S}$ -kernel的概念，该概念扩展了$b$ -kernel的概念。然而，这个定义不允许我们概括$b$ -正则序列的所有许多特征。本文给出了$\mathcal{S}$ -核的另一种定义，从而给出了$\mathcal{S}$ -正则序列的另一种定义，它使我们能够使用可识别的形式级数，以便将大多数（如果不是全部的话）已知的$b$ -正则序列的刻画推广到抽象的数制系统.然后给出了$\mathcal{S}$ -自动序列作为特殊$\mathcal{S}$ -正则序列的两个刻画.接下来，我们提出了一个通用的方法来获得各种家庭的$\mathcal{S}$ -正则序列通过列举$\mathcal{S}$ -可识别的属性的$\mathcal{S}$ -自动序列。作为这种方法的许多可能的应用程序的一个例子，我们表明，除了是$\mathcal{S}$ -可识别的，一个$\mathcal{S}$ -自动序列的因子复杂性定义了一个$\mathcal{S}$ -正则序列。在论文的最后一部分，我们研究了$\mathcal{S}$ -同步序列。一路上，我们证明了作为同步关系和可识别序列的组成而获得的形式序列是可识别的。因此，一个$\mathcal{S}$ -同步序列和一个$\mathcal{S}$ -正则序列的组成被证明是$\mathcal{S}$ -规则的。所有结果都是在任意维数$d$和任意半环$\mathbb{K}$上得到的.
英文摘要：
The notion of $b$-regular sequences was generalized to abstract numeration systems by Maes and Rigo in 2002. Their definition is based on a notion of $\mathcal{S}$-kernel that extends that of $b$-kernel. However, this definition does not allow us to generalize all of the many characterizations of $b$-regular sequences. In this paper, we present an alternative definition of $\mathcal{S}$-kernel, and hence an alternative definition of $\mathcal{S}$-regular sequences, which enables us to use recognizable formal series in order to generalize most (if not all) known characterizations of $b$-regular sequences to abstract numeration systems. We then give two characterizations of $\mathcal{S}$-automatic sequences as particular $\mathcal{S}$-regular sequences. Next, we present a general method to obtain various families of $\mathcal{S}$-regular sequences by enumerating $\mathcal{S}$-recognizable properties of $\mathcal{S}$-automatic sequences. As an example of the many possible applications of this method, we show that, provided that addition is $\mathcal{S}$-recognizable, the factor complexity of an $\mathcal{S}$-automatic sequence defines an $\mathcal{S}$-regular sequence. In the last part of the paper, we study $\mathcal{S}$-synchronized sequences. Along the way, we prove that the formal series obtained as the composition of a synchronized relation and a recognizable series is recognizable. As a consequence, the composition of an $\mathcal{S}$-synchronized sequence and a $\mathcal{S}$-regular sequence is shown to be $\mathcal{S}$-regular. All our results are presented in an arbitrary dimension $d$ and for an arbitrary semiring $\mathbb{K}$.
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