Above and below subgroups of a latticeordered group
Authors:
Richard N. Ball, Paul Conrad and Michael Darnel
Journal:
Trans. Amer. Math. Soc. 297 (1986), 140
MSC:
Primary 06F15; Secondary 20E22
MathSciNet review:
849464
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Abstract: In an group , this paper defines an subgroup to be above an subgroup (or to be below ) if for every integer , , and , . It is shown that for every subgroup , there exists an subgroup maximal below which is closed, convex, and, if the group is normalvalued, unique, and that for every subgroup there exists an subgroup maximal above which is saturated: if and , then . Given groups and , it is shown that every lattice ordering of the splitting extension , which extends the lattice orders of and and makes lie above , is uniquely determined by a lattice homomorphism from the lattice of principal convex subgroups of into the cardinal summands of . This extension is sufficiently general to encompass the cardinal sum of two groups, the lex extension of an group by an group, and the permutation wreath product of two groups. Finally, a characterization is given of those abelian groups that split off below: whenever is a convex subgroup of an group , is then a splitting extension of by for any subgroup maximal above in .
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 R. N. Ball, Convergence and Cauchy structures on lattice ordered groups, Trans. Amer. Math. Soc. 259 (1980), 357392. MR 567085 (81m:06039)
 [2]
 , The generalized orthocompletion and strongly projectable hull of a lattice ordered group, Canad. J. Math. 34 (1982), 621661. MR 663307 (84e:06021)
 [3]
 , The structure of the completion of a lattice ordered group, Pacific J. Math. (submitted).
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 , Topological lattice ordered groups, Pacific J. Math. 83 (1979), 126. MR 555035 (82h:06021)
 [5]
 A. Bigard, Contribution à la théorie des groupes réticulés, Thèse sci. math., Paris, 1969. MR 0250950 (40:4181)
 [6]
 A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math., vol. 608, SpringerVerlag, Berlin and New York, 1977. MR 0552653 (58:27688)
 [7]
 J. W. Brewer, P. Conrad and P. Montgomery, Lattice ordered groups and a conjecture for adequate domains, Proc. Amer. Math. Soc. 43 (1974), 3135. MR 0332616 (48:10942)
 [8]
 R. D. Byrd, Lattice ordered groups, Thesis, Tulane University, 1966.
 [9]
 P. Conrad, Lattice ordered groups, Lecture Notes, Tulane University, 1970.
 [10]
 , Lex subgroups of lattice ordered groups, Czechoslovak Math. J. 18 (1968), 86103. MR 0225697 (37:1290)
 [11]
 , The structure of an group that is determined by its minimal prime subgroups, Ordered Groups, Lecture Notes in Pure and Appl. Math., vol. 62, Dekker, New York, 1980.
 [12]
 P. Conrad, J. Harvey and C. Holland, The Hahn embedding theorem for lattice ordered groups, Trans. Amer. Math. Soc. 108 (1963), 143169. MR 0151534 (27:1519)
 [13]
 A. Glass, Ordered permutation groups, London Math. Soc. Lecture Notes Series 55, Cambridge Univ. Press, London, 1981. MR 645351 (83j:06004)
 [14]
 S. McCleary, Closed cls of a normal valued group...
 [15]
 E. Scrimger, A large class of small varieties of lattice ordered groups, Proc. Amer. Math. Soc. 51 (1975), 301306. MR 0384644 (52:5517)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608494647
PII:
S 00029947(1986)08494647
Article copyright:
© Copyright 1986
American Mathematical Society
