Representations of distributive semilattices in ideal lattices of   various algebraic structures
  We study the relationships among existing results about representations of distributive semilattices by ideals in dimension groups, von Neumann regular rings, C*-algebras, and complemented modular lattices. We prove additional representation results which exhibit further connections with the scattered literature on these different topics. 
K. R. Goodearl and F. Wehrung