Semilattices of groups and nonstable K-theory of extended Cuntz limits

We give an elementary characterization of those abelian monoidsM that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0}. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of all subgroups of M (viewed as a semigroup), and it makes it pos- sible to express M as a certain submonoid of a direct product ×G, where is a distributive semilattice with zero and G is an abelian group. When applied to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our results yield a full description of V (A) for C*-inductive limits A of finite sums of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or corners of O1 by projections, thus extending to the case including O1 earlier work by the authors together with K.R. Goodearl

An Algebraic Theory for Data Linkage

There are countless sources of data available to governments, companies, and citizens, which can be combined for good or evil. We analyse the concepts of combining data from common sources and linking data from different sources. We model the data and its information content to be found in a single source by an ordered partial monoid, and the transfer of information between sources by different types of morphisms. To capture the linkage between a family of sources, we use a form of Grothendieck construction to create an ordered partial monoid that brings together the global data of the family in a single structure. We apply our approach to database theory and axiomatic structures in approximate reasoning. Thus, ordered partial monoids provide a foundation for the algebraic study for information gathering in its most primitive form

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

SIMULTANEOUS REPRESENTATIONS OF SEMILATTICES BY LATTICES WITH PERMUTABLE CONGRUENCES

International audienceThe Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilatticeS is isomorphic to the semilattice Conc L of compact congruences of a lattice L. While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that $|S|\leq\aleph_1$. Furthermore, one can then take L with permutable congruences. This contrasts with the case where $|S| \geq\aleph_2$, where there are counterexamples S for which L cannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well. We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Conc functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square 2^2 can be lifted with respect to the Conc functor. We prove that the latter results cannot be extended to the cube, 2^3. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Conc functor, by lattices with permutable congruences. We also extend many of our results to lattices with almost permutable congruences, that is, \ga\jj\gb=\ga\gb\uu\gb\ga, for all congruences a and b. We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Conc functor on finite Boolean semilattices