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Vector lattices over subfields of the reals

Part of: Ordered structures

Published online by Cambridge University Press:  09 April 2009

P. Bixler ,
P. Conrad ,
W. B. Powell  and
C. Tsinakis
P. Bixler
Affiliation:
Department of Computer Science Virginia Polytechnic InstituteBlacksburg, Virginia 24061, U.S.A.
P. Conrad
Affiliation:
University of KansasLawrence, Kansas 66044, U.S.A.
W. B. Powell
Affiliation:
Oklahoma State UniversityStillwater, Oklahoma 74078, U.S.A.
C. Tsinakis
Affiliation:
Vanderbilt UniversityNashville, Tennessee 37235, U.S.A.
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Abstract

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In this paper we consider classes of vector lattices over subfields of the real numbers. Among other properties we relate the archimedean condition of such a vector lattice to the uniqueness of scalar multiplication and the linearity of l-automorphisms. If a vector lattice in the classes considered admits an essential subgroup that is not a minimal prime, then it also admits a non-linear l-automorphism and more than one scalar multiplication. It is also shown that each l-group contains a largest archimedean convex l-subgroup which admits a unique scalar multiplication.

MSC classification

Secondary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 359 - 375
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Anderson, M. and Conrad, P., ‘Epicomplete l-groups’, Algebra Universalis 12 (1981), 224–241.CrossRefGoogle Scholar
[2]Bleier, R., ‘Minimal vector lattice covers’, Bull. Austral. Math. Soc. 5 (1971), 331–335.CrossRefGoogle Scholar
[3]Conrad, P., Harvey, J., and Holland, C., ‘The Hahn embedding theorem for abelian lattice ordered groups’, Trans. Amer. Math. Soc. 108 (1963), 143–169.CrossRefGoogle Scholar
[4]Conrad, P. and Diem, J., ‘Polar preserving endomorphisms’, Illinois J. Math. 15 (1971), 222–240.Google Scholar
[5]Conrad, P., Lattice ordered groups, (Tulane Lecture Notes, 1970).Google Scholar
[6]Conrad, P., ‘Minimal vector lattice covers’, Bull. Austral. Math. Soc. 4 (1971), 35–59.CrossRefGoogle Scholar
[7]Conrad, P., ‘The hulls of representable l-groups and f-rings’, J. Austral. Math. Soc. 16 (1973), 385–415.CrossRefGoogle Scholar
[8]Conrad, P., ‘Changing the scalar multiplication on a vector lattice’, J. Austral. Math. Soc. 20 (1975), 332–347.CrossRefGoogle Scholar
[9]Conrad, P., The structure of an l-group that is determined by its minimal prime subgroups, pp. 1–21 (Proc. Boise State Conference on Ordered Groups, Marcel Dekker, 1980).Google Scholar
[10]Martinez, J., ‘Free products of abelian l-groups’, Czechoslovak. Math. J. 23 (1973), 349–361.CrossRefGoogle Scholar
[11]Martinez, J., ‘Torsion theory for lattice ordered groups’, Czechoslovak. Math. J. 25 (1975), 284–299.CrossRefGoogle Scholar
[12]Powell, W. B. and Tsinakis, C., ‘Free products in the class of abelian l-groups’, Pacific J. Math. 104 (1983), 429–442.CrossRefGoogle Scholar