Strong perforation in infinitely generated K0-groups of simple C-algebras

Let (G,G+) be an ordered abelian group. We say that G has strong perforation if there exists x ∈ G, x / ∈ G+, such that nx ∈ G+, nx 6= 0 for some natural number n. Otherwise, the group is said to be weakly unperforated. Examples of simple C-algebras whose ordered K0-groups have this property and for which the entire order structure on K0 is known have, until now, been restricted to the case where K0 is group isomorphic to the integers. We construct simple, separable, unital C-algebras with strongly perforated K0-groups group isomorphic to an arbitrary infinitely generated subgroup of the rationals, and determine the order structure on K0 in each case.