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The hyper-archimedean kernel sequence of a lattice-ordered group

Published online by Cambridge University Press:  17 April 2009

Jorge Martinez
Jorge Martinez
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida, USA.
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The hyper-archimedean kernel Ar(G) of a lattice-ordered group (hence forth l–group) is the largest hyper-archimedean convex l–subgroup of the l–group G. One defines Arσ (G), for an ordinal σ as if a is a limit ordinal, and as the unique l–ideal with the property that Arσ(G)/Ar.σ–1(G) = Ar(G/Arσ–1(G)), otherwise. The resulting "Loewy"-like sequence of characteristic l–ideals, Ar(G) ⊆ Ar2(G) ⊆ … ⊆ Arσ (G) ⊆ …, is called the hyper-archimedean kernel sequence. The first result of this note says that each Arσ(G) ⊆ Ar(G)”.

Most of the paper concentrates on archimedean l–groups; in particular, the hyper-archimedean kernels are identified for: D(X), where X is a Stone space, a large class of free products of abelian l–groups, and certain l–subrings of a product of real groups.

It is shown that even for archimedean l–groups the hyper-archimedean kernel sequence may proceed past Ar(G).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 3 , June 1974 , pp. 337 - 349
Copyright
Copyright © Australian Mathematical Society 1974

References

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