  For every partially ordered sets I, having a finite cofinal subset, and every
field K we build a unital, locally matricial and hence unit-regular K-algebra
B(I) such that the lattice of all its ideals is order isomorphic to the lattice
of all lower subsets of I. We show that the Grothendieck group of B(I), with
its natural partial order, is order isomorphic to the restricted Hahn power of
Z by I; this gives a contribution to solve the Realization Problem for
Dimension Groups with order-unit. Finally we show that the algebra B(I) has the
following features: (a) B(I) is prime if and only if I is lower directed; (b)
B(I) is primitive if and only if I has a coinitial chain; (c) B(I) is
semiartinian if and only if I is artinian, in which the case I is order
isomorphic to the primitive spectrum of B(I).
