Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T01:10:09.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions

Published online by Cambridge University Press:  20 November 2018

F. Dashiell ,
A. Hager  and
M. Henriksen
F. Dashiell
Affiliation:
R & D Associates, Marina Del Rey, California
A. Hager
Affiliation:
Wesleyan University, Middletown, Connecticut
M. Henriksen
Affiliation:
Harvey Mudd College, Claremont, California
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 paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified, for compact spaces X, in [6]. This condition is that every dense cozero set S in X should be C*-embedded in X (that is, all bounded continuous functions on S extend to X). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the F-spaces of [12]).

In Section 1, the notion of a completion with respect to sequential order convergence is first described in the setting of a commutative lattice group G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 657 - 685
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bernau, S. J., Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math Soc, (3). 15 (1965), 599–631.Google Scholar
2. Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Pub. 25 (Amer. Math. Soc, Providence, Rhode Island, Third Edition, Second Printing, 1973).Google Scholar
3. Blair, R. and Hager, A., Extensions of zero-sets and of real-valued functions, Math. Zeit. 136 (1974), 41–52.Google Scholar
4. Comfort, W. and Negrepontis, S., Continuous pseudometrics, Lecture notes in pure and applied mathematics 14 (Marcel Dekker Inc., New York, 1975).Google Scholar
5. Comfort, W., Hindman, N. and Negrepontis, S., F-spaces and their product with P-spaces, Pacific J. Math. 28 (1969), 489–502.Google Scholar
6. Dashiell, F., Non-weakly compact operators from order-Cauchy complete C(S) lattices, with applications to Baire classes, Trans. Amer. Math. Soc, to appear.Google Scholar
7. Everett, C. J., Sequence completion of lattice moduls, Duke Math. J. 11 (1944), 109–119.Google Scholar
8. Fuchs, L.. Partially ordered algebraic systems (Pergammon Press, London, 1963).Google Scholar
9. Fine, N. J., Gillman, L., and Lambek, J., Rings of quotients of rings of functions (McGill University Press, Montreal, 1965).Google Scholar
10. Gillman, L., A P-space and an extremally disconnected space whose product is not an F-space, Archiv. der Math. 11 (1960), 53–55.Google Scholar
11. Gleason, A., Projective topological spaces, 111. J. Math. 2 (1958), 482–489.Google Scholar
12. Gillman, L. and Henriksen, M., Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. 82 (1956), 366–391.Google Scholar
13. Gillman, L. and Jerison, M., Rings of continuous functions (D. Van Nostrand Co. Inc., Princeton, N.J., 1960).Google Scholar
14. Hager, A. W., On inverse-closed subalgebras of C(X), Proc. London Math. Soc. 19 (1969), 233–257.Google Scholar
15. Hahn, H., Réelle Funktionen, Leipzig, 1932, reprint (Chelsea, New York, 1948).Google Scholar
16. Heider, L., Compactifications of dimension zero, Proc. Amer. Math. Soc. 10 (1959), 377–384.Google Scholar
17. Johnson, D. G., Completion of an Archimedean f-ring, J. London Math. Soc. 40 (1965), 493–496.Google Scholar
18. Kohls, C., Hereditary properties of some special spaces, Arch. Math. (Basel. 12 (1961), 129–133.Google Scholar
19. Levy, R., Almost-P-spaces, Can. J. Math. 29 (1977), 284–288.Google Scholar
20. Luxemberg, W. and Zaanen, A., Riesz spaces (Amsterdam, 1971).Google Scholar
21. Mack, J. E. and Johnson, D. G., The Dedekind completion of C﹛X), Pac. J. Math. 20 (1967), 231–243.Google Scholar
22. Negrepontis, S., On the product of F-spaces, Trans. Amer. Math. Soc. 136 (1969), 339–346.Google Scholar
23. Papangelou, F., Order convergence and topological completion of commutative latticegroups, Math. Annale. 155 (1964), 81–107.Google Scholar
24. Quinn, J., Intermediate Riesz spaces, Pac. J. Math. 56 (1975), 225–263.Google Scholar
25. Walker, R., The Stone-Cech compactification, Ergebnisse der Mathematik und ihre Grenzgebeite 83 (Springer-Verlag, New York, 1974).Google Scholar
26. Weinberg, E., Higher degrees of distributivity in lattices of continuous functions, Thesis, Purdue University (1961).Google Scholar
27. Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (D. Van Nostrand, Princeton, 1958).Google Scholar