A \(K_{0}\)-avoiding dimension group with an order-unit of index two (Q855718)

The main result of the paper is that there exists a dimension group \(G\) whose positive cone is not isomorphic to the dimension monoid Dim\((L)\) of any lattice \(L\). A negative answer is provided to a question posed by the author whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since \(G\) has an order-unit of index 2, this also solves negatively a problem posed by \textit{K. R. Goodearl} about representability, with respect to \(K_0\), of dimension groups with order-unit of index 2 by unit-regular rings.