Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T01:39:04.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Free Vector Lattices

Published online by Cambridge University Press:  20 November 2018

Kirby A. Baker
Kirby A. Baker*
Affiliation:
Harvard University, Cambridge, Mass. and California Institute of Technology, Pasadena, Calif.
Rights & Permissions [Opens in a new window]

Extract

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.

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 58 - 66
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

*

The author is indebted to Professor Birkhoff for suggesting this application of his theorem. A similar application appears in Henrikson-Isbell (7, § 4).

References

1. Birkhoff, G., On the structure of abstract algebras, Proc. Cambridge Philos. Soc, 31 (1935), 433–454.Google Scholar
2. Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloq. Pub., 25, 3rd éd. (Amer. Math. Soc, Providence, 1967).Google Scholar
3. Brignole, D. and Ribeiro, H., On the universal equivalence for ordered abelian groups, Algebra & Logika Sem., Novosibirsk, 4 (1965), Issue 2, 51–55.Google Scholar
4. Dilworth, R. P., Structure and decomposition theory of lattices, Proc. Symposia in Pure Math., Vol. II (Amer. Math. Soc, Providence, 1961).10.1090/pspum/002/0125037CrossRefGoogle Scholar
5. Fuchs, L., Partially ordered algebraic systems (Addison-Wesley, Reading, Mass., 1963).Google Scholar
6. Gurevich, Yu. Sh. and Kokorin, A., Universal equivalence of ordered abelian groups, Algebra & Logika Sem., Novosibirsk, 2 (1963), Issue 1, 37–39.Google Scholar
7. Henrikson, M. and Isbell, J., Lattice-ordered rings and function rings, Pacific J. Math., 12 1962), 533–565.Google Scholar
8. Horn, A., On sentences which are true of direct unions of algebras, J. Symb. Logic, 16 (1955), 14–21.Google Scholar
9. Lyndon, R. C., Properties preserved under algebraic constructions, Bull. Amer. Math. Soc, 65 (1959), 287–299.Google Scholar
10. Mayer-Kalkschmidt, J. and Steiner, E., Some theorems in set theory and applications in the ideal theory of partially ordered sets, Duke Math. J., 31 (1964), 287–290.Google Scholar
11. Ribiero, H., The notion of universal completeness, Portugal. Math., 15 (1956), 83–86.Google Scholar
12. Ross, K. A. and Stone, A. H., Products of separable spaces, Amer. Math. Monthly, 71 (1964), 398–403.Google Scholar
13. Topping, D., Some homological pathology in vector lattices, Can. J. Math., 17 (1965), 411–428.Google Scholar
14. Weinberg, E. C., Free lattice-ordered abelian groups, Math. Ann., 151 (1963), 187–199.Google Scholar
15. Weinberg, E. C., Free lattice-ordered abelian groups II, Math. Ann., 159 (1965), 217–222.Google Scholar
16. Yosida, K., On vector lattice with a unit, Proc. Japan Acad. Tokyo, 17 (1941), 121–124.Google Scholar