In a lattice-ordered group G
a (descending) doubling chain is a sequence a1> a2>⋯> an>⋯ of positive
elements of G such that an≧ 2an+1. An element 0 < s ∈ G is singular if 0 ≦ g ≦ s
implies that g ∧ (s − g) = 0. The main theorems are as follows: 1. The following
two statements are equivalent: (a) every doubling chain in G is finite; (b)
G =⋃τ<αGτ(τ ranging over all ordinals less than some α), where Gτ is
an l-ideal of G, σ < τ implies that Gσ⊆ Gτ and Gτ+1∕Gτ is generated
by its singular elements, (i.e. a Specker group, à la Conrad). 2. If G is
hyper-archimedean as well then either of the above conditions is equivalent to:
(c) G is hyper-𝒵, i.e. every totally ordered l-homomorphic image of G is
cyclic.
