  We investigate categorical and amalgamation properties of the functor Idc
assigning to every partially ordered abelian group G its semilattice of compact
ideals Idc G. Our main result is the following. Theorem 1. Every diagram of
finite Boolean semilattices indexed by a finite dismantlable partially ordered
set can be lifted, with respect to the Idc functor, by a diagram of
pseudo-simplicial vector spaces. Pseudo-simplicial vector spaces are a special
kind of finite-dimensional partially ordered vector spaces (over the rationals)
with interpolation. The methods introduced make it also possible to prove the
following ring-theoretical result. Theorem 2. For any countable distributive
join-semilattices S and T and any field K, any (v,0)-homomorphism $f: S\to T$
can be lifted, with respect to the Idc functor on rings, by a homomorphism $f:
A\to B$ of K-algebras, for countably dimensional locally matricial algebras A
and B over K. We also state a lattice-theoretical analogue of Theorem 2 (with
respect to the Conc functor, and we provide counterexamples to various related
statements. In particular, we prove that the result of Theorem 1 cannot be
achieved with simplicial vector spaces alone.

General Mathematics;K-Theory and Homology;Rings and Algebras