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)

 
 

 

Varieties of automorphism groups of orders


Author: W. Charles Holland
Journal: Trans. Amer. Math. Soc. 288 (1985), 755-763
MSC: Primary 06F15; Secondary 08B20, 20B27, 20E10, 20F16
DOI: https://doi.org/10.1090/S0002-9947-1985-0776402-7
MathSciNet review: 776402
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The group $ A(\Omega )$ of automorphisms of a totally ordered set $ \Omega $ must generate either the variety of all groups or the solvable variety of class $ n$. In the former case, $ A(\Omega )$ contains a free group of rank $ {2^{{\aleph _0}}}$; in the latter case, $ A(\Omega )$ contains a free solvable group of class $ n - 1$ and rank $ {2^{{\aleph _0}}}$.


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

  • [1] C. C. Chang and A. Ehrenfeucht, A characterization of abelian groups of automorphisms of a simply ordering relation, Fund. Math. 51 (1962), 141-147. MR 0142629 (26:198)
  • [2] A. M. W. Glass, Ordered permutation groups, London Math. Soc. Lecture Notes Ser., Vol. 55, Cambridge Univ. Press, London and New York, 1981. MR 645351 (83j:06004)
  • [3] A. M. W. Glass, Y. Gurevich, W. C. Holland and S. Shelah, Rigid homogeneous chains, Math. Proc. Cambridge Philos. Soc. 89 (1981), 7-17. MR 591966 (82c:06001)
  • [4] A. M. W. Glass, W. C. Holland and S. H. McCleary, The structure of $ l$-group varieties, Algebra Universalis 10 (1980), 1-20. MR 552151 (81k:06026a)
  • [5] W. C. Holland, The largest proper variety of lattice ordered groups, Proc. Amer. Math. Soc. 57 (1976), 25-28. MR 0406902 (53:10688)
  • [6] J. Martinez, Varieties of lattice-ordered groups, Math. Z. 137 (1974), 256-284. MR 0354483 (50:6961)
  • [7] S. H. McCleary, The structure of intransitive ordered permutation groups, Algebra Universalis 6 (1976), 229-255. MR 0424638 (54:12597)
  • [8] J. Mycielski, Almost every function is independent, Fund. Math. 81 (1973), 43-48. MR 0339091 (49:3854)
  • [9] H. Neumann, Varieties of groups, Ergeb. Math. Grenzgeb., No. 37, Springer-Verlag, 1967. MR 0215899 (35:6734)
  • [10] T. Ohkuma, Sur quelques ensembles ordonnés linéarment, Fund. Math. 43 (1955), 326-337. MR 0084486 (18:868c)
  • [11] J. A. Read, Wreath products of non-overlapping lattice-ordered groups, Canad. Math. Bull. 17 (1975), 713-722. MR 0384642 (52:5515)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 06F15, 08B20, 20B27, 20E10, 20F16

Retrieve articles in all journals with MSC: 06F15, 08B20, 20B27, 20E10, 20F16


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0776402-7
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society