Geodesic
On the Cardinality of the Family of Precomplete Classes in~$P_E$
Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 864-869.

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

Let $E$ be an infinite set of cardinality $\mathbf m$, and let $P_E$ be the set of all functions defined on $E$. We prove that the cardinality of the family of all classes precomplete in $P_E$ is equal to $2^{2^{\mathbf m}}$. If $C_{\mathbb R}$ is the set of all continuous functions of real variables, then the cardinality of the family of all classes precomplete in $C_{\mathbb R}$ is equal to $2^{2^{\aleph_0}}$. 
@article{MZM_2005_78_6_a5,
     author = {S. S. Marchenkov},
     title = {On the {Cardinality} of the {Family} of {Precomplete} {Classes} in~$P_E$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {864--869},
     publisher = {mathdoc},
     volume = {78},
     number = {6},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a5/}
}
TY  - JOUR
AU  - S. S. Marchenkov
TI  - On the Cardinality of the Family of Precomplete Classes in~$P_E$
JO  - Matematičeskie zametki
PY  - 2005
SP  - 864
EP  - 869
VL  - 78
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a5/
LA  - ru
ID  - MZM_2005_78_6_a5
ER  - 
%0 Journal Article
%A S. S. Marchenkov
%T On the Cardinality of the Family of Precomplete Classes in~$P_E$
%J Matematičeskie zametki
%D 2005
%P 864-869
%V 78
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a5/
%G ru
%F MZM_2005_78_6_a5
S. S. Marchenkov. On the Cardinality of the Family of Precomplete Classes in~$P_E$. Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 864-869. http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a5/

[1] Yablonskii S. V., Vvedenie v diskretnuyu matematiku, Nauka, M., 1986 | MR

[2] Post E. L., “Introduction to a general theory of elementary propositions”, Amer. J. Math., 43 (1921), 163–185 | DOI | MR

[3] Post E. L., Two-Valued Iterative Systems of Mathematical Logic, Ann. Math. Studies, 5, Princeton Univ. Press, Princeton, 1941 | MR | Zbl

[4] Kuznetsov A. V., “O problemakh tozhdestva i funktsionalnoi polnoty algebraicheskikh sistem”, Trudy 3-go Vsesoyuznogo matematicheskogo s'ezda, T. 2, Izd-vo AN SSSR, M., 1956, 145–146

[5] Yablonskii S. V., “O funktsionalnoi polnote v trekhznachnom ischislenii”, Dokl. AN SSSR, 95:6 (1954), 1152–1156 | MR

[6] Yablonskii S. V., “Funktsionalnye postroeniya v $k$-znachnoi logike”, Tr. MIAN, 51, Nauka, M., 1958, 5–142

[7] Zakharova E. Yu., Kudryavtsev V. B., Yablonskii S. V., “O predpolnykh klassakh v $k$-znachnykh logikakh”, Dokl. AN SSSR, 186:3 (1969), 509–512 | Zbl

[8] Gavrilov G. P., “O funktsionalnoi polnote v schetnoznachnoi logike”, Problemy kibernetiki, no. 15, Nauka, M., 1965, 5–64

[9] Marchenkov S. S., “O moschnosti mnozhestva predpolnykh klassov v nekotorykh klassakh funktsii schetnoznachnoi logiki”, Problemy kibernetiki, no. 38, Nauka, M., 1981, 109–116

[10] Marchenkov S. S., “O klassakh Slupetskogo dlya determinirovannykh funktsii”, Diskretnaya matem., 10:2 (1998), 128–136 | Zbl

[11] Marchenkov S. S., “Funktsionalnye aspekty problemy polnoty dlya nekotorykh klassov avtomatnykh funktsii”, Diskretnaya matem., 12:2 (2000), 103–117 | Zbl

[12] Maltsev A. I., Algebraicheskie sistemy, Nauka, M., 1970

[13] Fried E., Pixley A. F., “The dual discriminator function in universal algebra”, Acta. Sci. Math., 41:1–2 (1979), 83–100 | Zbl