Archimedean Closed Lattice-Ordered Groups

Abstract

We show that, if an abelian lattice-ordered group is archimedean closed, then each principal l-ideal is also archimedean closed. This has given a positive answer to the question raised in 1965 and hence proved that the class of abelian archimedean closed lattice-ordered groups is a radical class. We also provide some conditions for latticeordered group F (∆, R) to be the unique archimedean closure of ∑ (∆, R). Introduction. Throughout, let G be a lattice-ordered group (lgroup). Let Γ be a root system, that is, Γ is a partially ordered set for which {α ∈ Γ | α ≥ γ} is totally ordered, for any γ ∈ Γ. Let {Hγ | γ ∈ Γ} be a collection of abelian totally-ordered groups indexed by Γ. V (Γ, Hγ) is the set of all functions v on Γ for which v(γ) ∈ Hγ and the support of each v satisfies ascending chain condition. V (Γ, Hγ) is an abelian group under addition. Furthermore, if we define an element of V (Γ, Hγ) to be positive, if it is positive at each maximal element of its support, then V (Γ, Hγ) is an abelian l-group, which we call a Hahn group on Γ. ∑ (Γ, Hγ) is the l-subgroup of V (Γ, Hγ) whose elements have finite supports. A root in a root system Γ is a totally ordered subset of Γ. F (Γ, Hγ) is the l-subgroup of V (Γ, Hγ) such that the support of each element is contained in a finite number of roots in Γ. A convex l-subgroup which is maximal with respect to not containing some g ∈ G is called regular and is a value of g. Element g is special if it has a unique value, and in this case the value is called a special value. A convex l-subgroup P of G is prime if a∧b = 0 in G implies that either a ∈ P or b ∈ P . Regular subgroups of G are prime and form a root system under inclusion, written Γ(G). A subset ∆ ⊆ Γ(G) is plenary if ∩∆ = {0} and ∆ is a dual ideal in Γ(G); that is, if δ ∈ ∆, γ ∈ Γ(G) and γ > δ, then γ ∈ ∆. If G is an abelian l-group, then G is l-isomorphic to Received by the editors on September 8, 1998, and in revised form on May 31, 2001. Copyright c ©2004 Rocky Mountain Mathematics Consortium