Geodesic
Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions
Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 657-685

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Dashiell, F.; Hager, A.; Henriksen, M. Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions. Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 657-685. doi: 10.4153/CJM-1980-052-0
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[1] 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] 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] 3. Blair, R. and Hager, A., Extensions of zero-sets and of real-valued functions, Math. Zeit. 136 (1974), 41–52. Google Scholar

[4] 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] 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] 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] 7. Everett, C. J., Sequence completion of lattice moduls, Duke Math. J. 11 (1944), 109–119. Google Scholar

[8] 8. Fuchs, L.. Partially ordered algebraic systems (Pergammon Press, London, 1963). Google Scholar

[9] 9. Fine, N. J., Gillman, L., and Lambek, J., Rings of quotients of rings of functions (McGill University Press, Montreal, 1965). Google Scholar

[10] 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] 11. Gleason, A., Projective topological spaces, 111. J. Math. 2 (1958), 482–489. Google Scholar

[12] 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] 13. Gillman, L. and Jerison, M., Rings of continuous functions (D. Van Nostrand Co. Inc., Princeton, N.J., 1960). Google Scholar

[14] 14. Hager, A. W., On inverse-closed subalgebras of C(X), Proc. London Math. Soc. 19 (1969), 233–257. Google Scholar

[15] 15. Hahn, H., Réelle Funktionen, Leipzig, 1932, reprint (Chelsea, New York, 1948). Google Scholar

[16] 16. Heider, L., Compactifications of dimension zero, Proc. Amer. Math. Soc. 10 (1959), 377–384. Google Scholar

[17] 17. Johnson, D. G., Completion of an Archimedean f-ring, J. London Math. Soc. 40 (1965), 493–496. Google Scholar

[18] 18. Kohls, C., Hereditary properties of some special spaces, Arch. Math. (Basel. 12 (1961), 129–133. Google Scholar

[19] 19. Levy, R., Almost-P-spaces, Can. J. Math. 29 (1977), 284–288. Google Scholar

[20] 20. Luxemberg, W. and Zaanen, A., Riesz spaces (Amsterdam, 1971). Google Scholar

[21] 21. Mack, J. E. and Johnson, D. G., The Dedekind completion of C﹛X), Pac. J. Math. 20 (1967), 231–243. Google Scholar

[22] 22. Negrepontis, S., On the product of F-spaces, Trans. Amer. Math. Soc. 136 (1969), 339–346. Google Scholar

[23] 23. Papangelou, F., Order convergence and topological completion of commutative latticegroups, Math. Annale. 155 (1964), 81–107. Google Scholar

[24] 24. Quinn, J., Intermediate Riesz spaces, Pac. J. Math. 56 (1975), 225–263. Google Scholar

[25] 25. Walker, R., The Stone-Cech compactification, Ergebnisse der Mathematik und ihre Grenzgebeite 83 (Springer-Verlag, New York, 1974). Google Scholar

[26] 26. Weinberg, E., Higher degrees of distributivity in lattices of continuous functions, Thesis, Purdue University (1961). Google Scholar

[27] 27. Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (D. Van Nostrand, Princeton, 1958). Google Scholar

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