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A \(K_{0}\)-avoiding dimension group with an order-unit of index two. (English) Zbl 1126.19001

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 K. R. Goodearl about representability, with respect to \(K_0\), of dimension groups with order-unit of index 2 by unit-regular rings.

MSC:

19A49 \(K_0\) of other rings
16E20 Grothendieck groups, \(K\)-theory, etc.
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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