Abstract. We compute the monoid V (L K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L K (E) and the lattice of order-ideals of V (L K (E)). When K is the field C of complex numbers, the algebra L C (E) is a dense subalgebra of the graph C * -algebra C * (E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

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We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions.

We study the Borel reducibility of isomorphism relations in the generalized Baire space κ κ . In the main result we show for inaccessible κ, that if T is a classifiable theory and T ′ is stable with OCP, then the isomorphism of models of T is Borel reducible to the isomorphism of models of T ′ .