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Throughout this paper
po-group will mean partially ordered abelian group. A subgroup H of a po-group G is
an o-ideal if H is a convex, directed subgroup of G. A subgroup M of G is a
value of 0≠g ∈ G if M is an o-ideal of G that is maximal without g. Let
ℳ(g) = {M ⊆ G∣M is a value of g} and ℳ∗(g) = ∩ℳ(g). Two positive elements
a,b ∈ G are pseudo disjoint (p-disjoint) if a ∈ℳ∗(b) and b ∈ℳ∗(a), and G is a
pseudo-lattice ordered group (pl-group) if each g ∈ G can be written g = a−b where
a and b are p-disjoint.
The main result of §2 shows that every pl-group G is a Riesz group. That is, G is
semiclosed (ng ≧ 0 implies g ≧ 0 for all g ∈ G and all positive integers n), and G
satisfies the Riesz interpolation property; if, whenever x1,⋯,xm, y1,⋯,yn are
elements of G and xi ≦ yj for 1 ≦ i ≦ m, 1 ≦ j ≦ n, then there is an element z ∈ G
such that xi ≦ z ≦ yj.
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