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Transactions of the American Mathematical Society

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



An even better representation for free lattice-ordered groups

Author: Stephen H. McCleary
Journal: Trans. Amer. Math. Soc. 290 (1985), 81-100
MSC: Primary 06F15
MathSciNet review: 787956
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Abstract: The free lattice-ordered group $ {F_\eta }$ (of rank $ \eta $) has been studied in two ways: via the Conrad representation on the various right orderings of the free group $ {G_\eta }$ (sharpened by Kopytov's observation that some one right ordering must by itself give a faithful representation), and via the Glass-McCleary representation as a pathologically $ o{\text{-}}2$-transitive $ l$-permutation group. Each kind of representation yields some results which cannot be obtained from the other. Here we construct a representation giving the best of both worlds--a right ordering $ ({G_\eta }, \leqslant )$ on which the action of $ {F_\eta }$ is both faithful and pathologically $ o{\text{-}}2$-transitive. This $ ({G_\eta }, \leqslant )$ has no proper convex subgroups. The construction is explicit enough that variations of it can be utilized to get a great deal of information about the root system $ {\mathcal{P}_\eta }$ of prime subgroups of $ {F_\eta }$. All $ {\mathcal{P}_\eta }$'s with $ 1 < \eta < \infty $ are $ o$-isomorphic. This common root system $ {\mathcal{P}_f}$ has only four kinds of branches (singleton, three-element, $ {\mathcal{P}_f}$ and $ {\mathcal{P}_{{\omega _0}}}$), each of which occurs $ {2^{{\omega _0}}}$ times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of $ {\mathcal{P}_f}$.

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Keywords: Free lattice-ordered group, ordered permutation group, right ordered group
Article copyright: © Copyright 1985 American Mathematical Society