Robustness of Pisot-regular sequences

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.