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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Embedding theorems and generalized discrete ordered abelian groups


Authors: Paul Hill and Joe L. Mott
Journal: Trans. Amer. Math. Soc. 175 (1973), 283-297
MSC: Primary 06A60
MathSciNet review: 0311540
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Abstract: Let G be a totally ordered commutative group. For each nonzero element $ g \in G$, let $ L(g)$ denote the largest convex subgroup of G not containing g. Denote by $ U(g)$ the smallest convex subgroup of G that contains g. The group G is said to be generalized discrete if $ U(g)/L(g)$ is order isomorphic to the additive group of integers for all $ g \ne 0$ in G. This paper is principally concerned with the structure of generalized discrete groups. In particular, the problem of embedding a generalized discrete group in the lexicographic product of its components, $ U(g)/L(g)$, is studied. We prove that such an embedding is not always possible (contrary to statements in the literature). However, we do establish the validity of this embedding when G is countable. In case F is o-separable as well as countable, the structure of G is completely determined.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0311540-6
PII: S 0002-9947(1973)0311540-6
Keywords: Totally ordered group, discrete group, generalized discrete, Hahn's embedding theorem, regular group
Article copyright: © Copyright 1973 American Mathematical Society