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Media type:
E-Article
Title:
An Even Better Representation for Free Lattice-Ordered Groups
Contributor:
McCleary, Stephen H.
Published:
American Mathematical Society, 1985
Published in:
Transactions of the American Mathematical Society, 290 (1985) 1, Seite 81-100
Language:
English
ISSN:
0002-9947
Origination:
Footnote:
Description:
The free lattice-ordered group Fη(of rank η) has been studied in two ways: via the Conrad representation on the various right orderings of the free group Gη(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-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 (Gη, ⩽) on which the action of Fηis both faithful and pathologically o-2-transitive. This (Gη, ⩽) 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 Pηof prime subgroups of Fη. All Pn's with$1 < \eta < \infty$are o-isomorphic. This common root system Pfhas only four kinds of branches (singleton, three-element, Pf, and Pω0 ) each of which occurs 2ω0 times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of Pf.