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A \\emph{generalized number system over $\\mathcal{O}$} (GNS for short) is a pair $(p,\\mathcal{D})$ where $p \\in \\mathcal{O}[x]$ is monic and $\\mathcal{D}\\subset\\mathcal{O}$ is a complete residue system modulo $p(0)$ containing $0$. If each $a \\in \\mathcal{O}[x]$ admits a representation of the form $a \\equiv \\sum_{j =0}^{\\ell-1} d_j x^j \\pmod{p}$ with $\\ell\\in\\mathbb{N}$ and $d_0,\\ldots, d_{\\ell-1}\\in\\mathcal{D}$ then the GNS $(p,\\mathcal{D})$ is said to have the \\emph{finiteness property}. 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A \\emph{generalized number system over $\\mathcal{O}$} (GNS for short) is a pair $(p,\\mathcal{D})$ where $p \\in \\mathcal{O}[x]$ is monic and $\\mathcal{D}\\subset\\mathcal{O}$ is a complete residue system modulo $p(0)$ containing $0$. If each $a \\in \\mathcal{O}[x]$ admits a representation of the form $a \\equiv \\sum_{j =0}^{\\ell-1} d_j x^j \\pmod{p}$ with $\\ell\\in\\mathbb{N}$ and $d_0,\\ldots, d_{\\ell-1}\\in\\mathcal{D}$ then the GNS $(p,\\mathcal{D})$ is said to have the \\emph{finiteness property}. 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