Hostname: page-component-7f64f4797f-n5rmq Total loading time: 0 Render date: 2025-11-04T00:28:45.599Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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