Vol. 61, No. 2, 1975

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Doubling chains, singular elements and hyper-𝒵 l-groups

Jorge Martinez

Vol. 61 (1975), No. 2, 503–506

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.

Mathematical Subject Classification
Primary: 06A55, 06A55
Received: 30 December 1974
Published: 1 December 1975
Jorge Martinez