Geodesic
The semiring of continous $[0,1]$-valued functions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 4, pp. 53-82.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the properties of prime ideals in semirings of continuous functions with values in the unit interval $[0,1]$ on topological spaces. We describe maximal and pure ideals of such semirings. We study homomorphisms of semirings of continuous $[0,1]$-valued functions. In terms of semirings of functions we characterize some properties of compacta. We show that the theory of ideals in these semirings differs from the case of rings of continuous functions. 
@article{FPM_2012_17_4_a3,
     author = {E. M. Vechtomov and E. N. Lubyagina},
     title = {The semiring of continous $[0,1]$-valued functions},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {53--82},
     publisher = {mathdoc},
     volume = {17},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_4_a3/}
}
TY  - JOUR
AU  - E. M. Vechtomov
AU  - E. N. Lubyagina
TI  - The semiring of continous $[0,1]$-valued functions
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2012
SP  - 53
EP  - 82
VL  - 17
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2012_17_4_a3/
LA  - ru
ID  - FPM_2012_17_4_a3
ER  - 
%0 Journal Article
%A E. M. Vechtomov
%A E. N. Lubyagina
%T The semiring of continous $[0,1]$-valued functions
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2012
%P 53-82
%V 17
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2012_17_4_a3/
%G ru
%F FPM_2012_17_4_a3
E. M. Vechtomov; E. N. Lubyagina. The semiring of continous $[0,1]$-valued functions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 4, pp. 53-82. http://geodesic.mathdoc.fr/item/FPM_2012_17_4_a3/

[1] Varankina V. I., Vechtomov E. M., Semënova I. A., “Polukoltsa nepreryvnykh neotritsatelnykh funktsii: delimost, idealy kongruentsii”, Fundament. i prikl. mat., 4:2 (1998), 493–510 | MR | Zbl

[2] Vechtomov E. M., “Izomorfizmy multiplikativnykh polugrupp kolets nepreryvnykh funktsii”, Sib. mat. zhurn., 19:4 (1978), 759–771 | MR | Zbl

[3] Vechtomov E. M., “Voprosy opredelyaemosti topologicheskikh prostranstv algebraicheskimi sistemami nepreryvnykh funktsii”, Itogi nauki i tekhn. Ser. Algebra. Topol. Geom., 28, 1990, 3–46 | MR | Zbl

[4] Vechtomov E. M., Koltsa nepreryvnykh funktsii na topologicheskikh prostranstvakh. Izbrannye temy, Uch. posobie, MPGU, M., 1992 | MR

[5] Vechtomov E. M., “Distributivnye reshëtki, funktsionalno predstavimye tsepyami”, Fundament. i prikl. mat., 2:1 (1996), 93–102 | MR | Zbl

[6] Vechtomov E.M., “Polukoltsa nepreryvnykh otobrazhenii”, Vestn. Vyatsk. gos. guman. un-ta, 2004, no. 10, 57–64

[7] Vechtomov E. M., Chuprakov D. V., “Glavnye yadra polupolei nepreryvnykh polozhitelnykh funktsii”, Fundament. i prikl. mat., 14:4 (2008), 87–107 | MR

[8] Grettser G., Obschaya teoriya reshëtok, Mir, M., 1982 | MR

[9] Lubyagina E. N., “Polukoltsa nepreryvnykh funktsii so znacheniyami v edinichnom otrezke”, Mezhdunar. konf. “Maltsevskie chteniya”, Novosibirsk, 2009, 127

[10] Lubyagina E. N., “Maksimalnye idealy v polukoltsakh nepreryvnykh funktsii so znacheniyami v edinichnom otrezke”, Materialy Devyatoi molodëzhnoi nauchnoi shkoly-konferentsii “Lobachevskie chteniya – 2010”, Tr. Mat. tsentra im. N. I. Lobachevskogo, 40, 2010, 206–211

[11] Chermnykh V. V., Polukoltsa, Uch. posobie, Izd-vo VyatGGU, Kirov, 1997

[12] Chermnykh V. V., “Redutsirovannye rikkartovy polukoltsa i ikh funktsionalnye predstavleniya”, Fundament. i prikl. mat., 13:2 (2007), 205–215 | MR | Zbl

[13] Engelking R., Obschaya topologiya, Mir, M., 1986 | MR

[14] Araujo J., Multiplicative bijections of semigroups of interval-valued continuous functions, UC Davis Math., 2007, 11 pp. http://trefoil.math.ucdavis.edu/0710.4347

[15] Golan J. S., Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999 | MR

[16] Jerison M., Gilman L., Rings of Continuous Functions, Van Nostrand, Princeton, 1976

[17] Marovt J., “Multiplicative bijections of $C(X,\mathbf I)$”, Proc. Am. Math. Soc., 134 (2006), 1065–1075 | DOI | MR | Zbl

[18] Milgram A. N., “Multiplicative semigroups of continuous functions”, Duke Math. J., 16:2 (1949), 377–383 | DOI | MR | Zbl