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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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Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups
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by Stephen H. McCleary PDF
Trans. Amer. Math. Soc. 290 (1985), 69-79 Request permission

Abstract:

It is easy to pose questions about the free lattice-ordered group ${F_\eta }$ of rank $\eta > 1$ whose answers$^{2}$ are "obvious", but difficult to verify. For example: 1. What is the center of ${F_\eta }$? 2. Is ${F_\eta }$ directly indecomposable? 3. Does ${F_\eta }$ have a basic element? 4. Is ${F_\eta }$ completely distributive? Question 1 was answered recently by Medvedev, and both $1$ and $2$ by Arora and McCleary, using Conrad’s representation of ${F_\eta }$ via right orderings of the free group ${G_\eta }$. Here we answer all four questions by using a completely different tool: The (faithful) representation of ${F_\eta }$ as an $o{\text {-}}2$-transitive $l$-permutation group which is pathological (has no nonidentity element of bounded support). This representation was established by Glass for most infinite $\eta$, and is here extended to all $\eta > 1$. Curiously, the existence of a transitive representation for ${F_\eta }$ implies (by a result of Kopytov) that in the Conrad representation there is some right ordering of ${G_\eta }$ which suffices all by itself to give a faithful representation of ${F_\eta }$. For finite $\eta$, we find that every transitive representation of ${F_\eta }$ can be made from a pathologically $o{\text {-}}2$-transitive representation by blowing up the points to $o$-blocks; and every pathologically $o{\text {-}}2$-transitive representation of ${F_\eta }$ can be extended to a pathologically $o{\text {-}}2$-transitive representation of ${F_{{\omega _0}}}$.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 290 (1985), 69-79
  • MSC: Primary 06F15
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0787955-7
  • MathSciNet review: 787955