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

Abstract

Given an infinite cardinal ℵ, let $Po{s}_{\aleph }$ be the category whose objects are all partially ordered sets I, having a finite cofinal subset and such that both I and the set of all maximal chains of I have at most cardinality ℵ, while the morphisms are all maps which are order imbeddings of the domain as an upper subset of the codomain. Given a field $K$, for every object I of $Po{s}_{\aleph }$ we find a unital, locally matricial and hence unit-regular $K$-algebra $B_{\aleph ,K}\left(I\right)$ 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 $K_0\left(B_{\aleph ,K}\right(I\left)\right)$, 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. We also show that the algebra $B_{\aleph ,K}\left(I\right)$ has the following features: (a) $B_{\aleph ,K}\left(I\right)$ is prime if and only if I is lower directed; (b) $B_{\aleph ,K}\left(I\right)$ is primitive if and only if I has a coinitial chain; (c) $B_{\aleph ,K}\left(I\right)$ is semiartinian if and only if I is artinian, in which case I is order isomorphic to the primitive spectrum of $B_{\aleph ,K}\left(I\right)$. Finally we show that the assignment $I\mapsto B_{\aleph ,K}\left(I\right)$ extends to a pair of functors from $Po{s}_{\aleph }$ to the category of $K$-algebras, one covariant and the other contravariant.