Free lattice-ordered groups represented as - transitive -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 of rank whose answers are "obvious", but difficult to verify. For example: 1. What is the center of ? 2. Is directly indecomposable? 3. Does have a basic element? 4. Is completely distributive?

Question 1 was answered recently by Medvedev, and both and by Arora and McCleary, using Conrad's representation of via right orderings of the free group . Here we answer all four questions by using a completely different tool: The (faithful) representation of as an -transitive -permutation group which is pathological (has no nonidentity element of bounded support). This representation was established by Glass for most infinite , and is here extended to all . Curiously, the existence of a transitive representation for implies (by a result of Kopytov) that in the Conrad representation there is some right ordering of which suffices all by itself to give a faithful representation of . For finite , we find that every transitive representation of can be made from a pathologically -transitive representation by blowing up the points to -blocks; and every pathologically -transitive representation of can be extended to a pathologically -transitive representation of .

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0787955-7

Keywords:
Free lattice-ordered group,
ordered permutation group,
right ordered group

Article copyright:
© Copyright 1985
American Mathematical Society