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Transactions of the American Mathematical Society

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



Free lattice-ordered groups represented as $ o$-$ 2$ transitive $ l$-permutation groups

Author: Stephen H. McCleary
Journal: Trans. Amer. Math. Soc. 290 (1985), 69-79
MSC: Primary 06F15
MathSciNet review: 787955
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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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Keywords: Free lattice-ordered group, ordered permutation group, right ordered group
Article copyright: © Copyright 1985 American Mathematical Society