Geodesic
Maximal d-Ideals in a Riesz Space
Canadian journal of mathematics, Tome 35 (1983) no. 6, pp. 1010-1029

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

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). 
Huijsmans, Charles B.; Pagter, Ben de. Maximal d-Ideals in a Riesz Space. Canadian journal of mathematics, Tome 35 (1983) no. 6, pp. 1010-1029. doi: 10.4153/CJM-1983-056-6
@article{10_4153_CJM_1983_056_6,
     author = {Huijsmans, Charles B. and Pagter, Ben de},
     title = {Maximal {d-Ideals} in a {Riesz} {Space}},
     journal = {Canadian journal of mathematics},
     pages = {1010--1029},
     year = {1983},
     volume = {35},
     number = {6},
     doi = {10.4153/CJM-1983-056-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-056-6/}
}
TY  - JOUR
AU  - Huijsmans, Charles B.
AU  - Pagter, Ben de
TI  - Maximal d-Ideals in a Riesz Space
JO  - Canadian journal of mathematics
PY  - 1983
SP  - 1010
EP  - 1029
VL  - 35
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-056-6/
DO  - 10.4153/CJM-1983-056-6
ID  - 10_4153_CJM_1983_056_6
ER  - 
%0 Journal Article
%A Huijsmans, Charles B.
%A Pagter, Ben de
%T Maximal d-Ideals in a Riesz Space
%J Canadian journal of mathematics
%D 1983
%P 1010-1029
%V 35
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-056-6/
%R 10.4153/CJM-1983-056-6
%F 10_4153_CJM_1983_056_6

[1] 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] 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] 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] 4. Bernau, S. J., Topologies on structure spaces of lattice groups, Pac. J. Math. 42 (1972), 557–568. Google Scholar

[5] 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] 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] 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] 8. Fremlin, D. H., Riesz spaces with the order continuity property I, Proc. Camb. Phil. Soc. 81 (1977), 31–42. Google Scholar

[9] 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] 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] 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] 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] 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] 14. Huijsmans, C. B. and de Pagter, B., Ideal theory in f-algebras, Trans. Am. Math. Soc. 269 (1982), 225–245. Google Scholar

[15] 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] 16. Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I (North-Holland Publishing Company, Amsterdam-London, 1971). Google Scholar

[17] 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] 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] 19. Papangelou, F., Order convergence and topological completion of commutative latticegroups, Math. Ann. 755 (1964), 81–107. Google Scholar

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

[21] 21. Seever, G. L., Measures on F-spaces, Trans. Am. Math. Soc. 133 (1968), 267–280. Google Scholar

[22] 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] 23. Tucker, C. T., Concerning σ-homomorphisms of Riesz spaces, Pac. J. Math. 57 (1975), 585–590. Google Scholar

Cité par Sources :