Robustness of Pisot-regular sequences

Abstract: We consider numeration systems based on a $d$-tuple $\mathbf{U}=(U_1,\ldots,U_d)$ of sequences of integers and we define $(\mathbf{U},\mathbb{K})$-regular sequences through $\mathbb{K}$-recognizable formal series, where $\mathbb{K}$ is any semiring. We show that, for any $d$-tuple $\mathbf{U}$ of Pisot numeration systems and any commutative semiring $\mathbb{K}$, this definition does not depend on the greediness of the $\mathbf{U}$-representations of integers. The proof is constructive and is based on the fact that the normalization is realizable by a $2d$-tape finite automaton. In particular, we use an ad hoc operation mixing a $2d$-tape automaton and a $\mathbb{K}$-automaton in order to obtain a new $\mathbb{K}$-automaton.