Abstract :
We study the partially ordered abelian group, F(P,R), generated by a set of generators, P, and inequality relations, R, and its representations in Euclidean space. First F(P, R) is defined by a universal property and then existence is shown proof-theoretically. We then characterize the order structure of F(P,R). The first main result of the paper utilizes the Marriage Theorem to prove that if the set of relations, R, is derived from a finite partially ordered set then F(P, R) is isomorphically embeddable in Rn for n sufficiently large. The second main result utilizes the Compactness Theorem to prove that for finite R, F(P, R) is pseudo-Archimedean.
