Regular sequences and synchronized sequences in abstract numeration systems
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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 ๐ฎ-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 ๐ฎ-kernel, and hence an alternative definition of ๐ฎ-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 ๐ฎ-automatic sequences as particular ๐ฎ-regular sequences. Next, we present a general method to obtain various families of ๐ฎ-regular sequences by enumerating ๐ฎ-recognizable properties of ๐ฎ-automatic sequences. As an example of the many possible applications of this method, we show that, provided that addition is ๐ฎ-recognizable, the factor complexity of an ๐ฎ-automatic sequence defines an ๐ฎ-regular sequence. In the last part of the paper, we study ๐ฎ-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 ๐ฎ-synchronized sequence and a ๐ฎ-regular sequence is shown to be ๐ฎ-regular. All our results are presented in an arbitrary dimension d and for an arbitrary semiring ๐.
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