Ideal lattices of dimension groups
by
Pavel Ruzicka
Charles University in Prague

A dimension group is a partially ordered directed and unperforated abelian group with the interpolation property. Directed convex subgroups of a dimension group form an algebraic distributive lattice. We prove that not every algebraic distributive lattice is represented as the lattice of directed convex subgroups of a dimension group. The least cardinality of the semilattice of compact elements of such a lattice is aleph 1. Then we focus on the semilattice of compact elements of a given algebraic lattice. We will study the representation problem for semilattices obtained as a direct limit of a countable chain of distributive lattices and join-semilattice homomorphisms.

Date received: May 21, 2004