Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-14T14:25:28.649Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal d-Ideals in a Riesz Space

Published online by Cambridge University Press:  20 November 2018

Charles B. Huijsmans  and
Ben de Pagter
Charles B. Huijsmans
Affiliation:
Leiden State University, Leiden, The Netherlands'
Ben de Pagter
Affiliation:
California Institute of Technology, Pasadena, 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.

We recall that the ideal I in an Archimedean Riesz space L is called a d-ideal whenever it follows from ƒ ∊ I that {ƒ}dd ⊂ I. Several authors (see [4], [5], [6], [12], [13], [15] and [18]) have considered the class of all d-ideals in L, but the set ℐd of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact thatℐd, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L.

The main purpose of the present paper is to investigate the topological properties of ℐd and to compare ℐd to other structure spaces of L, such as the space of minimal prime ideals and the space of all e-maximal ideals in L (where e > 0 is a weak order unit).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1010 - 1029
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Aliprantis, C. D. and Langford, E., Almost σ-Dedekind complete Riesz spaces and the main inclusion theorem, Proc. Am. Math. Soc. 44 (1974), 421–426.Google Scholar
2. Amemiya, I., A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ., Ser. I 12 (1953), 111–156.Google Scholar
3. Aron, E. R. and Hager, A. W., Convex vector lattices and l-algebras, Topology and its Applications 72 (1981), 1–10.Google Scholar
4. Bernau, S. J., Topologies on structure spaces of lattice groups, Pac. J. Math. 42 (1972), 557–568.Google Scholar
5. Bigard, A., Keimel, K. and Wolfenstein, S., Groupes et anneaux reticules, Lecture Notes in Mathematics 608 (Springer Verlag, Berlin-Heidelberg-New York, 1977).Google Scholar
6. Bondarev, A. S., The presence of projections in quotient lineals of vector lattices, Dokl. Akad. Nauk. UzSSR. 8 (1974), 5–7.Google Scholar
7. Dashiell, F. K. Jr., Hager, A. W. and Henriksen, M., Order Cauchy completions of rings and vector lattices of continuous functions, Can. J. Math. 32 (1980) 657–685.Google Scholar
8. Fremlin, D. H., Riesz spaces with the order continuity property I, Proc. Camb. Phil. Soc. 81 (1977), 31–42.Google Scholar
9. Gillman, L. and Jerison, M., Rings of continuous functions, Graduate Texts in Math. 43 (Springer Verlag, Berlin-Heidelberg-New York, 1976).Google Scholar
10. Henriksen, M. and Jerison, J., The space of minimal prime ideals of a commutative ring, Trans. Am. Math. Soc. 115 (1965), 110–130.Google Scholar
11. Henriksen, M. and Johnson, D. G., On the structure of a class of Archimedean lattice-ordered algebras. Fund. Math. 50 (1961), 73–94.Google Scholar
12. Huijsmans, C. B. and de Pagter, B., On z-ideals and d-ideals in Riesz spaces I, Indag. Math. 42 (Proc. Neth. Acad. Sc. A83, 1980), 183–195.Google Scholar
13. Huijsmans, C. B. and de Pagter, B., On z-ideals and d-ideals in Riesz spaces II, Indag. Math. 42 (Proc. Neth. Acad. Sc. A83, 1980), 391–408.Google Scholar
14. Huijsmans, C. B. and de Pagter, B., Ideal theory in f-algebras, Trans. Am. Math. Soc. 269 (1982), 225–245.Google Scholar
15. Luxemburg, W. A. J., Extensions of prime ideals and the existence of projections in Riesz spaces, Indag. Math. 35 (Proc. Neth. Acad. Sc. A76, 1973), 263–279.Google Scholar
16. Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I (North-Holland Publishing Company, Amsterdam-London, 1971).Google Scholar
17. Meyer, M., Une nouvelle caractérisation des espaces vectoriels réticulés presque a-complets, C. R. Acad. Sc. Paris 287 (A)(1978), 1081–1084.Google Scholar
18. de Pagter, B., On z-ideals and d-ideals in Riesz spaces III, Indag. Math. 43 (Proc. Neth. Acad. Sc. A84, 1981), 409–422.Google Scholar
19. Papangelou, F., Order convergence and topological completion of commutative latticegroups, Math. Ann. 755 (1964), 81–107.Google Scholar
20. Quinn, J., Intermediate Riesz spaces, Pac. J. Math. 56 (1975), 225–263.Google Scholar
21. Seever, G. L., Measures on F-spaces, Trans. Am. Math. Soc. 133 (1968), 267–280.Google Scholar
22. Speed, T. P., Spaces of ideals in distributive lattices II, Minimal prime ideals, J. Austr. Math. Soc. 77 (1974), 54–72.Google Scholar
23. Tucker, C. T., Concerning σ-homomorphisms of Riesz spaces, Pac. J. Math. 57 (1975), 585–590.Google Scholar