Regular sequences in abstract numeration systems

An abstract numeration system S is given by a regular language L over a totally ordered alphabet (A,<). The numeration language L is ordered thanks to the radix (or genealogical) order induced by <. A natural number n is then represented by the n-th word of the language (where we start counting from n=0). Integer bases b and numeration systems based on a linear base sequence U with a regular numeration language are examples of abstract numeration systems. The notion of b-regular sequences was extended to abstract numeration systems by Maes and Rigo. 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 talk, I will present an alternative definition of S-kernel, and hence an alternative definition of S-regular sequences, that permits us to use recognizable formal series in order to generalize most (if not all) known characterizations of b-regular sequences. I will also show that an extra characterization can be obtained in the case of Pisot numeration systems. Finally, I will consider the special cases of S-automatic and S-synchronized sequences. In particular, we will see that the factor complexity of an S-automatic sequence defines an S-regular sequence. This is a joint work with Célia Cisternino and Manon Stipulanti.