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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Embedding theorems and generalized discrete ordered abelian groups
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by Paul Hill and Joe L. Mott PDF
Trans. Amer. Math. Soc. 175 (1973), 283-297 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 175 (1973), 283-297
  • MSC: Primary 06A60
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0311540-6
  • MathSciNet review: 0311540