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For every partially ordered sets I, having a finite cofinal subset, and everyfield K we build a unital, locally matricial and hence unit-regular K-algebraB(I) such that the lattice of all its ideals is order isomorphic to the latticeof all lower subsets of I. We show that the Grothendieck group of B(I), withits natural partial order, is order isomorphic to the restricted Hahn power ofZ by I; this gives a contribution to solve the Realization Problem forDimension Groups with order-unit. Finally we show that the algebra B(I) has thefollowing 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) issemiartinian if and only if I is artinian, in which the case I is orderisomorphic to the primitive spectrum of B(I).

Representation of partially ordered sets over Von Neumann regular algebras. More prime, non-primitive regular rings