Geodesic
Algebras and spaces of dense constancies
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 449-461 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

A DC-space (or space of dense constancies) is a Tychonoff space $X$ such that for each $f\in C(X)$ there is a family of open sets $\lbrace U_i\: i\in I\rbrace $, the union of which is dense in $X$, such that $f$, restricted to each $U_i$, is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean $f$-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions are dense), and it is shown that all metrizable spaces have this property.
A DC-space (or space of dense constancies) is a Tychonoff space $X$ such that for each $f\in C(X)$ there is a family of open sets $\lbrace U_i\: i\in I\rbrace $, the union of which is dense in $X$, such that $f$, restricted to each $U_i$, is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean $f$-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions are dense), and it is shown that all metrizable spaces have this property.
Classification : 06F25, 16S90, 54C05, 54C35, 54G99
Keywords: space and algebra of dense constancy; $c$-spectrum 
@article{CMJ_2001_51_3_a0,
     author = {Bella, A. and Martinez, J. and Woodward, S. D.},
     title = {Algebras and spaces of dense constancies},
     journal = {Czechoslovak Mathematical Journal},
     pages = {449--461},
     year = {2001},
     volume = {51},
     number = {3},
     mrnumber = {1851539},
     zbl = {1079.54506},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a0/}
}
TY  - JOUR
AU  - Bella, A.
AU  - Martinez, J.
AU  - Woodward, S. D.
TI  - Algebras and spaces of dense constancies
JO  - Czechoslovak Mathematical Journal
PY  - 2001
SP  - 449
EP  - 461
VL  - 51
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a0/
LA  - en
ID  - CMJ_2001_51_3_a0
ER  - 
%0 Journal Article
%A Bella, A.
%A Martinez, J.
%A Woodward, S. D.
%T Algebras and spaces of dense constancies
%J Czechoslovak Mathematical Journal
%D 2001
%P 449-461
%V 51
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a0/
%G en
%F CMJ_2001_51_3_a0
Bella, A.; Martinez, J.; Woodward, S. D. Algebras and spaces of dense constancies. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 449-461. http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a0/

[1] F. W.  Anderson: Lattice-ordered rings of quotients. Canad. J.  Math. 17 (1965), 434–448. | DOI | MR | Zbl

[2] M.  Anderson and T.  Feil: Lattice-ordered groups. An introduction. Reidel Texts in the Math. Sciences, Kluwer, Dordrecht, 1988. | MR

[3] B.  Banaschewski: Maximal rings of quotients of semi-simple commutative rings. Arch. Math. 16 (1965), 414–420. | DOI | MR | Zbl

[4] A.  Bella, A. W.  Hager, J.  Martinez, S.  Woodward and H.  Zhou: Specker spaces and their absolutes, I. Topology Appl. 72 (1996), 259–271. | DOI | MR

[5] A.  Bigard, K.  Keimel, S.  Wolfenstein: Groupes et Anneaux Réticulés. LNM, Vol.  608. Springer-Verlag, Berlin-Heidelberg-New York, 1977. | MR

[6] R.  Bleier: The orthocompletion of a lattice-ordered group. Indag. Math. 38 (1976), 1–7. | DOI | MR | Zbl

[7] L.  Gillman, M.  Jerison: Rings of Continuous Functions. GTM Vol.  43. Springer-Verlag, Berlin-Heidelberg-New York, 1976. | MR

[8] A.  Hager: Minimal covers of topological spaces. Ann. New York Acad. Sci.; Papers on general topology and related category theory and topological algebra Alg. Vol. 552 (1989), 44–59. | MR | Zbl

[9] J.  Lambek: Lectures on Rings and Modules. Blaisdell Publ. Co., Waltham, 1966. | MR | Zbl

[10] J.  Martinez: The maximal ring of quotients of an $f$-ring. Algebra Universalis 33 (1995), 355–369. | DOI | MR

[11] J.  Martinez and S.  Woodward: Specker spaces and their absolutes, II. Algebra Universalis 35 (1996), 333–341. | DOI | MR

[12] J.  Porter, R. G.  Woods: Extensions and Absolutes of Hausdorff Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1988. | MR

[13] Y.  Utumi: On quotient rings. Osaka Math.  J. 8 (1956), 1–18. | MR | Zbl