Regular sequences and synchronized sequences in abstract numeration systems

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 \(\mathbb{K}\).