Characterization of matrix types of ultramatricial algebras

A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every group element is smaller than a multiple of u. For every subgroup H of the multiplicatice groups of the positive rational numbers a dimension group is constructed whose order-preserving automorphism group is H and H acts on an order unit by multiplication as rational numbers. This implies that for every equivalence relation on the positive integers for which n and m are equivalent if and only if nk and mk are equivalent, there is a ring over which the ring of n times n matrices and the ring of m times m matrices are equivalent if and only if n and m are equivalent.