Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1985-0787956-9
MathSciNet review: 787956
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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}$.


References [Enhancements On Off] (What's this?)

  • [1] A. Kumar Arora and S. H. McCleary, Centralizers in free lattice-ordered groups, Houston J. Math. (to appear). MR 873641 (88a:06021)
  • [2] P. Conrad, Free lattice-ordered groups, J. Algebra 16 (1970), 191-203. MR 0270992 (42:5875)
  • [3] A. M. W. Glass, Ordered permutation groups, London Math. Soc. Lecture Note Ser. 55, Cambridge Univ. Press, London and New York, 1981. MR 645351 (83j:06004)
  • [4] V. M. Kopytov, Free lattice-ordered groups, Algebra and Logic 18 (1979), 259-270. (English translation) MR 582096 (81i:06018)
  • [5] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin and New York, 1976. MR 1812024 (2001i:20064)
  • [6] S. H. McCleary, $ O{\text{-}}2$-transitive ordered permutation groups, Pacific J. Math 49 (1973), 425-429. MR 0349525 (50:2018)
  • [7] -, Free lattice-ordered groups represented as $ o{\text{-}}2$-transitive $ l$-permutation groups, Trans. Amer. Math. Soc. 290 (1985), 69-79. MR 787955 (86m:06034a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 06F15

Retrieve articles in all journals with MSC: 06F15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0787956-9
Keywords: Free lattice-ordered group, ordered permutation group, right ordered group
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society