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A root system Λ is a partially
ordered set having the property that no two incomparable elements λ and μ have a
common lower bound. Π(Λ,Rλ) will denote the direct product of copies of R, the set
of real numbers, one for each λ ∈ Λ.V (Λ,Rλ) is the following subgroup:
v ∈ V = V (Λ,Rλ) if the support of v has no infinite ascending sequences. We put a
lattice order on v by setting v ≧ 0 if v = 0 or else every maximal component of v is
positive in R.
This paper has two main results: we first show that the cone of any finite
dimensional vector lattice G can be obtained as the union of an increasing sequence
P1,P2⋯ of archimedean vector lattice cones on G such that (G,P1)≅(G,P2)≅⋯ , as
vector lattices. Next, generalizing this, we show that for any root system Λ the cone
of the l-group V = V (Λ,Rλ) can be obtained as the union of a family of archimedean
vector l-cones {Pγ : γ ∈ Γ} on V , where (V,Pγ)≅(V,Pδ), as vector lattices, for all
γ,δ ∈ Γ.
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