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