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

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

MathSciNet review:
787955

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Abstract | References | Similar Articles | Additional Information

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 .

**[1]**Ashok Kumar Arora and Stephen H. McCleary,*Centralizers in free lattice-ordered groups*, Houston J. Math.**12**(1986), no. 4, 455–482. MR**873641****[2]**S. J. Bernau,*Free abelian lattice groups*, Math. Ann.**180**(1969), 48–59. MR**0241340**, https://doi.org/10.1007/BF01350085**[3]**Paul Conrad,*Free lattice-ordered groups*, J. Algebra**16**(1970), 191–203. MR**0270992**, https://doi.org/10.1016/0021-8693(70)90024-4**[4]**Paul F. Conrad,*Free abelian 𝑙-groups and vector lattices*, Math. Ann.**190**(1971), 306–312. MR**0281667**, https://doi.org/10.1007/BF01431159**[5]**Paul Conrad and Donald McAlister,*The completion of a lattice ordered group*, J. Austral. Math. Soc.**9**(1969), 182–208. MR**0249340****[6]**A. M. W. Glass,*𝑙-simple lattice-ordered groups*, Proc. Edinburgh Math. Soc. (2)**19**(1974/75), no. 2, 133–138. MR**0409309**, https://doi.org/10.1017/S0013091500010257**[7]**A. M. W. Glass,*Ordered permutation groups*, Bowling Green State University, Bowling Green, Ohio, 1976. MR**0422105****[8]**A. M. W. Glass,*Ordered permutation groups*, London Mathematical Society Lecture Note Series, vol. 55, Cambridge University Press, Cambridge-New York, 1981. MR**645351****[9]**W. Charles Holland and Stephen H. McCleary,*Solvability of the word problem in free lattice-ordered groups*, Houston J. Math.**5**(1979), no. 1, 99–105. MR**533643****[10]**V. M. Kopytov,*Free lattice-ordered groups*, Algebra i Logika**18**(1979), no. 4, 426–441, 508 (Russian). MR**582096****[11]**Stephen H. McCleary,*Closed subgroups of lattice-ordered permutation groups*, Trans. Amer. Math. Soc.**173**(1972), 303–314. MR**0311535**, https://doi.org/10.1090/S0002-9947-1972-0311535-1**[12]**Stephen H. McCleary,*𝑜-2-transitive ordered permutation groups*, Pacific J. Math.**49**(1973), 425–429. MR**0349525****[13]**-,*Free lattice-ordered 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), 691-696 (English translation).

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Additional Information

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