Coding of the dimension group

We study an aspect of dimension group theory, linked to coding. The dimension group that we consider is built on a given square primitive integer matrix M   satisfying the conditions that |detM|⩾2|detM|⩾2 and that the characteristic polynomial of M   is irreducible. The coding is based on iteration of what could be seen as a generalization to ZdZd of the Euclidean algorithm induced by the matrix M and in a natural way we define a binary operation of addition in the coding group.The set B   of symbols is a subset of ZdZd, and if we denote by ρ the Perron–Frobenius eigenvalue of M and by v a left eigenvector associated to ρ  , we define a function Zd×BN*→RZd×BN*→R which assigns to the element (p,b1,b2,…)(p,b1,b2,…) the series〈v,p〉+1ρ〈v,b1〉+1ρ2〈v,b2〉+⋯ (in case M=(10)M=(10), this is the decimal expansion) and the restriction of this function to finite codes is the classical embedding of the dimension group into RR.Finally, and under some suitable conditions, we prove that the last function is surjective and this allows the coding of real numbers and consequently the dimension group embedded into RR appears as the set of decimal numbers.