Vol. 36, No. 2, 1971

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Approximation by archimedean lattice cones

Jorge Martinez

Vol. 36 (1971), No. 2, 427–437

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 ,Rλ) is the following subgroup: v V = V ,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 ,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.

Mathematical Subject Classification
Primary: 06.30
Secondary: 46.00
Received: 3 February 1970
Published: 1 February 1971
Jorge Martinez