Free latticeordered groups represented as  transitive permutation groups
Author:
Stephen H. McCleary
Journal:
Trans. Amer. Math. Soc. 290 (1985), 6979
MSC:
Primary 06F15
MathSciNet review:
787955
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Abstract: It is easy to pose questions about the free latticeordered 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 .
 [1]
Ashok
Kumar Arora and Stephen
H. McCleary, Centralizers in free latticeordered groups,
Houston J. Math. 12 (1986), no. 4, 455–482. MR 873641
(88a:06021)
 [2]
S.
J. Bernau, Free abelian lattice groups, Math. Ann.
180 (1969), 48–59. MR 0241340
(39 #2680)
 [3]
Paul
Conrad, Free latticeordered groups, J. Algebra
16 (1970), 191–203. MR 0270992
(42 #5875)
 [4]
Paul
F. Conrad, Free abelian 𝑙groups and vector lattices,
Math. Ann. 190 (1971), 306–312. MR 0281667
(43 #7382)
 [5]
Paul
Conrad and Donald
McAlister, The completion of a lattice ordered group, J.
Austral. Math. Soc. 9 (1969), 182–208. MR 0249340
(40 #2585)
 [6]
A.
M. W. Glass, 𝑙simple latticeordered groups, Proc.
Edinburgh Math. Soc. (2) 19 (1974/75), no. 2,
133–138. MR 0409309
(53 #13069)
 [7]
A.
M. W. Glass, Ordered permutation groups, Bowling Green State
University, Bowling Green, Ohio, 1976. MR 0422105
(54 #10097)
 [8]
A.
M. W. Glass, Ordered permutation groups, London Mathematical
Society Lecture Note Series, vol. 55, Cambridge University Press,
CambridgeNew York, 1981. MR 645351
(83j:06004)
 [9]
W.
Charles Holland and Stephen
H. McCleary, Solvability of the word problem in free
latticeordered groups, Houston J. Math. 5 (1979),
no. 1, 99–105. MR 533643
(80f:06018)
 [10]
V.
M. Kopytov, Free latticeordered groups, Algebra i Logika
18 (1979), no. 4, 426–441, 508 (Russian). MR 582096
(81i:06018)
 [11]
Stephen
H. McCleary, Closed subgroups of latticeordered
permutation groups, Trans. Amer. Math. Soc.
173 (1972),
303–314. MR 0311535
(47 #97), http://dx.doi.org/10.1090/S00029947197203115351
 [12]
Stephen
H. McCleary, 𝑜2transitive ordered permutation
groups, Pacific J. Math. 49 (1973), 425–429. MR 0349525
(50 #2018)
 [13]
, Free latticeordered groups, Ordered Algebraic Structures (Proc. 1982 Special Session on Ordered Groups, Cincinnati, Ohio), Dekker, New York, 1984.
 [14]
N. Ya. Medvedev, Decomposition of free groups into direct products, Siberian Math. J. 21 (1981), 691696 (English translation).
 [1]
 A. Kumar Arora and S. H. McCleary, Centralizers in free latticeordered groups, Houston J.Math. (to appear). MR 873641 (88a:06021)
 [2]
 S. J. Bernau, Free abelian lattice groups, Math. Ann. 180 (1969), 4859. MR 0241340 (39:2680)
 [3]
 P. Conrad, Free latticeordered groups, J. Algebra 16 (1970), 191203. MR 0270992 (42:5875)
 [4]
 , Free abelian groups and vector lattices, Math. Ann. 190 (1971), 306312. MR 0281667 (43:7382)
 [5]
 P. Conrad and D. McAlister, The completion of a latticeordered group, J. Austral. Math. Soc. 9 (1969), 182208. MR 0249340 (40:2585)
 [6]
 A. M. W. Glass, simple latticeordered groups, Proc. Edinburgh Math. Soc. (2) 19 (1974), 133138. MR 0409309 (53:13069)
 [7]
 , Ordered permutation groups, Bowling Green State Univ., Bowling Green, Ohio, 1976. MR 0422105 (54:10097)
 [8]
 , Ordered permutation groups, London Math. Soc. Lecture Note Ser. 55, Cambridge Univ. Press, London and New York, 1981. MR 645351 (83j:06004)
 [9]
 W. C. Holland and S. H. McCleary, Solvability of the word problem in free latticeordered groups, Houston J. Math. 5 (1979), 99105. MR 533643 (80f:06018)
 [10]
 V. M. Kopytov, Free latticeordered groups, Algebra and Logic 18 (1979), 259270 (English translation). MR 582096 (81i:06018)
 [11]
 S. H. McCleary, Closed subgroups of latticeordered permutation groups, Trans. Amer. Math. Soc. 173 (1972), 303314. MR 0311535 (47:97)
 [12]
 , transitive ordered permutation groups, Pacific J. Math. 49 (1973), 425429. MR 0349525 (50:2018)
 [13]
 , Free latticeordered groups, Ordered Algebraic Structures (Proc. 1982 Special Session on Ordered Groups, Cincinnati, Ohio), Dekker, New York, 1984.
 [14]
 N. Ya. Medvedev, Decomposition of free groups into direct products, Siberian Math. J. 21 (1981), 691696 (English translation).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507879557
PII:
S 00029947(1985)07879557
Keywords:
Free latticeordered group,
ordered permutation group,
right ordered group
Article copyright:
© Copyright 1985
American Mathematical Society
