Functional representations of semirings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 3, pp. 111-227.

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

The theory of sheaf representations of semirings is developed in this paper.
@article{FPM_2012_17_3_a8,
     author = {V. V. Chermnykh},
     title = {Functional representations of semirings},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {111--227},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_3_a8/}
}
TY  - JOUR
AU  - V. V. Chermnykh
TI  - Functional representations of semirings
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2012
SP  - 111
EP  - 227
VL  - 17
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2012_17_3_a8/
LA  - ru
ID  - FPM_2012_17_3_a8
ER  - 
%0 Journal Article
%A V. V. Chermnykh
%T Functional representations of semirings
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2012
%P 111-227
%V 17
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2012_17_3_a8/
%G ru
%F FPM_2012_17_3_a8
V. V. Chermnykh. Functional representations of semirings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 3, pp. 111-227. http://geodesic.mathdoc.fr/item/FPM_2012_17_3_a8/

[1] Andrunakievich V. A., Ryabukhin Yu. M., Radikaly algebr i strukturnaya teoriya, Nauka, M., 1979 | MR

[2] Beidar K. I., Mikhalëv A. V., “Ortogonalnaya polnota i minimalnye pervichnye idealy”, Tr. seminara im. I. G. Petrovskogo, 10, 1984, 227–234 | MR | Zbl

[3] Beidar K. I., Mikhalëv A. V., “Ortogonalnaya polnota i algebraicheskie sistemy”, Uspekhi mat. nauk, 40:6 (1985), 79–115 | MR | Zbl

[4] Birkgof G., Teoriya reshëtok, Nauka, M., 1984 | MR

[5] Bredon G., Teoriya puchkov, Nauka, M., 1988 | MR | Zbl

[6] Bukur I., Delyanu A., Vvedenie v teoriyu kategorii i funktorov, Mir, M., 1972 | MR

[7] Varankina V. I., “Maksimalnye idealy v polukoltsakh nepreryvnykh funktsii”, Fundament. i prikl. mat., 1:4 (1995), 923–937 | MR | Zbl

[8] 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

[9] Vechtomov E. M., “Chistye idealy kolets i teorema Bkusha”, Abelevy gruppy i moduli, 8, 1989, 54–64 | MR

[10] Vechtomov E. M., Koltsa nepreryvnykh funktsii na topologicheskikh prostranstvakh. Izbrannye temy, Izd-vo MPGU, M., 1992 | MR

[11] Vechtomov E. M., “Annulyatornye kharakterizatsii bulevykh kolets i bulevykh reshëtok”, Mat. zametki, 53:2 (1993), 15–24 | MR | Zbl

[12] Vechtomov E. M., Koltsa nepreryvnykh funktsii so znacheniyami v topologicheskom tele, Dis. $\dots$ dokt. fiz.-mat. nauk, MPGU, M., 1993

[13] Vechtomov E. M., Funktsionalnye predstavleniya kolets, Izd-vo MPGU, M., 1993 | MR

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

[15] Vechtomov E. M., Vvedenie v polukoltsa, Izd-vo Vyatskogo gos. ped. un-ta, Kirov, 2000

[16] Vechtomov E. M., Mikhalëv A. V., Chermnykh V. V., “Abelevo regulyarnye polozhitelnye polukoltsa”, Tr. seminara im. I. G. Petrovskogo, 20, 1997, 282–309 | Zbl

[17] Vechtomov E. M., Cheraneva A. V., “Polutela i ikh svoistva”, Fundament. i prikl. mat., 14:5 (2008), 3–54 | MR

[18] Vechtomov E. M., Chermnykh V. V., “Usloviya simmetrichnosti v koltsakh i polukoltsakh”, Vestn. Vyatskogo gos. ped. un-ta. Matematika, informatika, fizika, 1996, no. 1, 6–8

[19] Gelfand I. M., “O normirovannykh koltsakh”, DAN SSSR, 22:1 (1939), 430–432

[20] Godeman R., Algebraicheskaya topologiya i teoriya puchkov, Izd. inostr. lit., M., 1961 | MR

[21] Goldblat R., Toposy. Kategornyi analiz logiki, Mir, M., 1983 | MR | Zbl

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

[23] Dzhonston P., Teoriya toposov, Nauka, M., 1986 | MR

[24] Kasivara M., Shapira P., Puchki na mnogoobraziyakh, Mir, M., 1997

[25] Kon P., Universalnaya algebra, Mir, M., 1968 | MR

[26] Kelli Dzh., Obschaya topologiya, Nauka, M., 1968

[27] Lambek I., Koltsa i moduli, Mir, M., 1971 | MR | Zbl

[28] Lyubetskii V. A., “Otsenki i puchki. O nekotorykh voprosakh nestandartnogo analiza”, Uspekhi mat. nauk, 44:4 (1989), 99–153 | MR | Zbl

[29] Maslov V. P., Kolokoltsov V. N., Idempotentnyi analiz i ego primenenie v optimalnom upravlenii, Nauka, M., 1994 | MR

[30] L. A. Skornyakov (red.), Obschaya algebra, v. 1, Nauka, M., 1990 ; т. 2, 1991 | Zbl

[31] Starostina O. V., “Stroenie abelevo-regulyarnykh polozhitelnykh polukolets”, Chebyshëvskii sb., 6:4(16) (2005), 142–151 | MR

[32] Tyukavkin D. V., Pirsovskie puchki dlya kolets s involyutsiei, Dep. v VINITI, No 3446–82 Dep., MGU, M., 1982

[33] Chermnykh V. V., “Predstavleniya polozhitelnykh polukolets secheniyami”, Uspekhi mat. nauk, 47:5 (1992), 193–194 | MR | Zbl

[34] Chermnykh V. V., “Puchkovye predstavleniya polukolets”, Uspekhi mat. nauk, 48:5 (1993), 185–186 | MR | Zbl

[35] Chermnykh V. V., Puchkovye predstavleniya polukolets, Dis. $\dots$ kand. fiz.-mat. nauk, MPGU, M., 1993

[36] Chermnykh V. V., “Lambekovskoe predstavlenie polukolets”, Vestn. Vyatskogo gos. ped. un-ta. Matematika, informatika, fizika, 1996, no. 1, 19–21 | MR

[37] Chermnykh V. V., “O polnote puchkovykh predstavlenii polukolets”, Fundament. i prikl. mat., 2:1 (1996), 267–277 | MR | Zbl

[38] Chermnykh V. V., “O predpuchke polukolets endomorfizmov”, Vestn. Vyatskogo gos. ped. un-ta. Matematika, informatika, fizika, 1997, no. 3, 33–36

[39] Chermnykh V. V., Polukoltsa, Izd-vo Vyatskogo gos. ped. un-ta, Kirov, 1997

[40] Chermnykh V. V., “Analog teoremy Stouna–Veiershtrassa dlya polukolets”, Nauch. vestn. Kirovskogo filiala MGEI, 1998, no. 1, 193–196

[41] Chermnykh V. V., “Lambekovskoe predstavlenie polumodulei”, Vestn. Vyatskogo gos. guman. un-ta, 2003, no. 8, 107–109

[42] Chermnykh V. V., “Gelfandovy polukoltsa i ikh predstavleniya secheniyami”, Chebyshëvskii sb., 5:2 (2004), 131–148 | MR | Zbl

[43] Chermnykh V. V., “Dvoistvennost Malvi dlya gelfandovykh polukolets”, Vestn. Vyatskogo gos. guman. un-ta, 2004, no. 10, 64–66

[44] Chermnykh V. V., “O regulyarnykh polukoltsakh s nekotorymi usloviyami”, Vestn. Vyatskogo gos. guman. un-ta, 2004, no. 11, 147–149

[45] Chermnykh V. V., “O polnom puchkovom predstavlenii polukolets”, Vestn. Vyatckogo gos. guman. un-ta. Informatika. Matematika. Yazyk, 2005, no. 3, 163–165

[46] Chermnykh V. V., “Polukoltsa sechenii puchkov”, Vestn. Vyatskogo gos. guman. un-ta, 2005, no. 13, 151–158

[47] Chermnykh V. V., “Predstavleniya polumodulei secheniyami puchkov”, Fundament. i prikl. mat., 13:2 (2007), 195–204 | MR | Zbl

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

[49] Chermnykh V. V., Funktsionalnye predstavleniya polukolets i polumodulei, Dis. $\dots$ dokt. fiz.-mat. nauk, IMM UrO RAN, Ekaterinburg, 2007

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

[51] Ahsan J., Latif R., “Representations of weakly regular semirings by sections in a presheaf”, Commun. Algebra, 21:8 (1993), 2819–2835 | DOI | MR | Zbl

[52] M. P. Fourman, C. J. Mulvey, D. S. Scott (eds.), Applications of Sheaves, Proc. Res. Symp. Appl. Sheaf Theory to Logic, Algebra and Anal. (Univ. Durham, Durham, 1977), Lect. Notes Math., 753, Springer, Berlin, 1979 | DOI | MR | Zbl

[53] Arens R., Kaplansky I., “Topological representations of algebras”, Trans. Am. Math. Soc., 63 (1949), 457–481 | DOI | MR

[54] Bkouche R., “Couples spectraux et faisceaux associes. Applications aux anneaux de fonctions”, Bull. Soc. Math. France, 98:3 (1970), 253–295 | MR | Zbl

[55] Borceux F., Simmons H., Van den Bossche G., “A sheaf representation for modules with applications to Gelfand rings”, Proc. London Math. Soc., 48:2 (1984), 230–246 | DOI | MR | Zbl

[56] Brezuleanu A., Diaconescu R., “Sur la duable de la categorie des treillis”, Rev. Roumaine Math. Pures Appl., 14:3 (1969), 311–323 | MR | Zbl

[57] Brown B., McCoy N. H., “Some theorems on groups with applications to ring theory”, Trans. Am. Math. Soc., 69 (1950), 302–311 | DOI | MR | Zbl

[58] Burgess W. D., Stephenson W., “Pierce sheaves of non-commutative rings”, Commun. Algebra, 39 (1976), 512–526 | DOI | MR

[59] Burgess W. D., Stephenson W., “An analogue of the Pierce sheaf for non-commutative rings”, Commun. Algebra, 6:9 (1978), 863–886 | DOI | MR | Zbl

[60] Burgess W. D., Stephenson W., “Rings all of whose Pierce stalks are local”, Can. Math. Bull., 22:2 (1979), 159–164 | DOI | MR | Zbl

[61] Carson A. B., “Representation of regular rings of finite index”, J. Algebra, 39:2 (1976), 512–526 | DOI | MR | Zbl

[62] Cignoli R., “The lattice of global sections of sheaves of chains over Boolean spaces”, Algebra Universalis, 8:3 (1978), 357–373 | DOI | MR | Zbl

[63] Comer S. D., “Representation by algebras of sections over Boolean spaces”, Pacific J. Math., 38 (1971), 29–38 | DOI | MR | Zbl

[64] Cornish W. H., “Normal lattices”, J. Aust. Math. Soc., 14:2 (1972), 200–215 | DOI | MR | Zbl

[65] Cornish W. H., “$0$-ideals, congruences and sheaf representations of distributive lattices”, Rev. Roumaine Math. Pures Appl., 22:8 (1977), 200–215 | MR

[66] Dauns J., Hofmann K. H., “The representation of biregular rings by sheaves”, Math. Z., 91:2 (1966), 103–123 | DOI | MR | Zbl

[67] Dauns J., Hofmann K. H., Representation of rings by sections, Mem. Amer. Math. Soc., 83, 1968, 180 pp. | MR | Zbl

[68] Davey B. A., “Sheaf spaces and sheaves of universal algebras”, Math. Z., 134:4 (1973), 275–290 | DOI | MR | Zbl

[69] Dedekind R., “Über die Theorie ganzen algebraischen Zahlen”, Suppl. XI to: P. G. Lejeune-Dirichlet, Vorlesungen über Zahlentheorie, Vieweg und Sohn, Braunschweig, 1894 | Zbl

[70] Filipoiu A., “Compact sheaves of lattices and normal lattices”, Math. Japon., 36:2 (1991), 381–386 | MR | Zbl

[71] Georgescu G., “Pierce representations of distributive lattices”, Kobe J. Math., 10:1 (1993), 1–11 | MR | Zbl

[72] Georgescu G., Voiculescu J., “Isomorphic sheaf representations of normal lattices”, J. Pure Appl. Algebra, 45:3 (1987), 213–223 | DOI | MR | Zbl

[73] Glazek K., A Short Guide Through the Literature on Semirings, Preprint No. 39, University of Wroclaw, Math. Inst., Wroclaw, 1985 | MR

[74] Glazek K., A Short Guide to the Literature on Semirings and Their Applications in Mathematics and Computer Science, Technical University Press, 2000 | MR | Zbl

[75] Golan J. S., Localization of Noncommutative Rings, Marcel Dekker, New York, 1975 | MR | Zbl

[76] Golan J. S., Structure sheaves over a noncommutative ring, Lect. Notes Pure Appl. Math., 56, Marcel Dekker, New York, 1980 | MR | Zbl

[77] Golan J. S., “Two sheaf constructions for noncommutative rings”, Houston J. Math., 6:1 (1980), 59–66 | MR | Zbl

[78] Golan J. S., The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Pitman Monogr. Surv. Pure Appl. Math., 54, Harlow, 1992 | MR | Zbl

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

[80] Grothendieck A., “Sur quelques points d'algebre homologique”, Tôhoku Math. J., 9 (1957), 119–221 | MR | Zbl

[81] Grothendieck A., Dieudonne J., Éléments de Géométrie Algébrique, v. 1, Publ. Math., 4, Inst. Hautes Études Sci., Paris, 1960 | DOI | Zbl

[82] Hebisch U., Weinert H. J., Semirings, Algebraic Theory Appl. Comp. Sci., World Scientific, Singapore, 1998 | MR | Zbl

[83] Hilbert D., “Über den Zahlbergriff”, Jahresber. Deutsch. Math. Verein, 8 (1899), 180–184

[84] Hofmann K. H., “Representations of algebras by continuons sections”, Bull. Am. Math. Soc., 78:3 (1972), 291–373 | DOI | MR | Zbl

[85] Huntington E. V., “Complete sets of postulates for the theory of positive integral and positive rational numbers”, Trans. Am. Math. Soc., 3 (1902), 280–284 | DOI | MR | Zbl

[86] Keimel K., “Darstellung von Halbgruppen und universellen Algebren durch Schnitte in Garben; biregulare Halbgruppen”, Math. Nachr., 45 (1970), 81–96 | DOI | MR | Zbl

[87] Keimel K., “The representation of lattice-ordered groups and rings by sections in sheaves”, Lectures Appl. Sheaves Ring Theory, Tulane Univ. Ring Operator Theory Year, 1970–1971, v. 3, Lect. Notes Math., 248, Springer, Berlin, 1971, 1–98 | DOI | MR

[88] Kennison J. F., “Integral domain type representations in sheaves and other topoi”, Math. Z., 151:1 (1976), 35–56 | DOI | MR | Zbl

[89] Koh K., “On functional representations of ring without nilpotent elements”, Can. Math. Bull., 14:3 (1971), 349–352 | DOI | MR | Zbl

[90] Koh K., “On a representation strongly harmonic ring by sheaves”, Pacific J. Math., 41:2 (1972), 459–468 | DOI | MR | Zbl

[91] Lambek J., “On representation of modules by sheaves of factor modules”, Can. Math. Bull., 14:3 (1971), 359–368 | DOI | MR | Zbl

[92] Macaulay F. S., Algebraic Theory of Modular Systems, Cambridge Univ. Press, Cambridge, 1916 | MR | Zbl

[93] MacLane S., “History of abstract algebra: origin, rise and decline of a movement”, Amer. Math. Heritage: Algebra and Applied Mathematics, Texas Univ., 1981, 11–35 | MR

[94] Mulvey C. J., “Intuitionistic algebra and representations of rings”, Mem. Amer. Math. Soc., 148 (1974), 3–57 | MR | Zbl

[95] Mulvey C. J., “Compact ringed spaces”, J. Algebra, 52:2 (1978), 411–436 | DOI | MR | Zbl

[96] Mulvey C. J., “A generalization of Gelfand duality”, J. Algebra, 56:2 (1979), 499–505 | DOI | MR | Zbl

[97] Mulvey C. J., “Representations of rings and modules”, Applications of Sheaves, Proc. Res. Symp. Appl. Sheaf Theory to Logic, Algebra and Anal. (Univ. Durham, Durham, 1977), Lect. Notes Math., 753, eds. M. P. Fourman, C. J. Mulvey, D. S. Scott, Springer, Berlin, 1979, 542–585 | DOI | MR

[98] Pierce R. S., “Modules over commutative regular rings”, Mem. Amer. Math. Soc., 70 (1967), 1–112 | MR

[99] Recent Advances in the Representation Theory of Rings and $\mathbb C^*$-Algebras by Continuous Sections, Mem. Amer. Math. Soc., 148, Amer. Math. Soc., 1974 | MR

[100] Rumbos I. B., “A structure sheaf for semiprime rings”, Commun. Algebra, 17:11 (1989), 2773–2794 | DOI | MR | Zbl

[101] Serre J. P., “Faisceaux algébriques cohérents”, Ann. Math. (2), 61 (1955), 197–278 | DOI | MR | Zbl

[102] Simmons H., “Reticulated rings”, J. Algebra, 64:1 (1980), 169–192 | DOI | MR

[103] Simmons H., “Sheaf representations of strongly harmonic rings”, Proc. Roy. Soc. Edinburgh. Sect. A, 99:3–4 (1985), 249–268 | DOI | MR | Zbl

[104] Simmons H., “Compact representations the lattice theory of compact ringed spaces”, J. Algebra, 126:2 (1989), 493–531 | DOI | MR | Zbl

[105] Stone M., “Applications of the theory of Boolean rings to general topology”, Trans. Am. Math. Soc., 41:3 (1937), 375–481 | DOI | MR | Zbl

[106] Teleman S., “Representation par faisceaux des modules sur les anneaux harmoniques”, C. R. Acad. Sci. Paris, 269:17 (1969), A753–A756 | MR

[107] Vandiver H. S., “Note on a simple type of algebras in which cancellation law of addition does not hold”, Bull. Am. Math. Soc., 40 (1934), 914–920 | DOI | MR | Zbl

[108] Vechtomov E. M., “Rings and sheaves”, J. Math. Sci., 74:1 (1995), 749–798 | DOI | MR | Zbl

[109] Werner H., “Sheaf constructions in universal algebra and model theory”, Universal Algebra and Applications, Pap. Stefan Banach Inst. Math. Cent. Semestr. Febr. 15 – June 9, 1978, Warsaw, 1982, 133–179 | MR | Zbl