Representation of algebraic distributive lattices with ℵ1\aleph_1 compact elements as ideal lattices of regular rings

We prove the following result: Theorem. Every algebraic distributive lattice $D$ with at most $\aleph _1$ compact elements is isomorphic to the ideal lattice of a von Neumann regular ring $R$. (By earlier results of the author, the $\aleph _1$ bound is optimal.) Therefore, $D$ is also isomorphic to the congruence lattice of a sectionally complemented modular lattice $L$, namely, the principal right ideal lattice of $R$. Furthermore, if the largest element of $D$ is compact, then one can assume that $R$ is unital, respectively, that $L$ has a largest element. This extends several known results of G. M. Bergman, A. P. Huhn, J. Tuma, and of a joint work of G. Grätzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10]. The main tool used in the proof of our result is an amalgamation theorem for semilattices and algebras (over a given division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. Tuma, and G. Grätzer, H. Lakser, and the author

Semilattices of groups and inductive limits of Cuntz algebras

We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form ðZ=nZÞ t f0g (where 0 is a new zero element), for positive integers n. The key properties are the Riesz refinement property and the requirement that each element x has finite order, that is, ðn þ 1Þx ¼ x for some positive integer n. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids in question as certain submonoids of direct products L G for semilattices L and torsion abelian groups G. When applied to the monoids VðAÞ appearing in the non-stable K-theory of C*-algebras, our results yield characterizations of the monoids VðAÞ for C* inductive limits A of sequences of finite direct products of matrix algebras over Cuntz algebras On. In particular, this completely solves the problem of determining the range of the invariant in the unital case of Rørdam’s classification of inductive limits of the above type

Monoids of intervals of simple refinement monoids and non-stable K-Theory of multiplier algebras

We show that the representation of the monoid of intervals of a simple refinement monoid in terms of affine semicontinuous functions, given by Perera in 2001, fails to be faithful in the case of strictly perforated monoids. We give some potential applications of this result in the context of monoids of intervals and K-Theory of multiplier rings

Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras

The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea

Primely generated refinement monoids

We extend both Dobbertin's characterization of primely generated regular refinement monoids and Pierce's characterization of primitive monoids to general primely generated refinement monoids.The first-named author was partially supported by DGI MINECO MTM2011-28992-C02-01, by FEDER UNAB10-4E-378 "Una manera de hacer Europa", and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The second-named author was partially supported by the DGI and European Regional Development Fund, jointly, through Project MTM2011-28992-C02-02, and by PAI III grants FQM-298 and P11-FQM-7156 of the Junta de Andalucía

Nonstable K-Theory for graph algebras

We compute the monoid V (LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(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 LK(E) and the lattice of order-ideals of V (LK(E)). When K is the field C of complex numbers, the algebra LC(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 propert