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Fundam. Prikl. Mat., 1998, Volume 4, Issue 2, Pages 493–510 (Mi fpm329)  

This article is cited in 19 scientific papers (total in 19 papers)

Semirings of continuous nonnegative functions: divisibility, ideals, congruences

V. I. Varankina, E. M. Vechtomov, I. A. Semenova

Vyatka State Pedagogical University

Abstract: Authors investigate the properties of divisibility (GCD, LCM, to be Bezout semiring) in semirings of continuous nonnegative real-valued functions on a topological space $X$. The correspondences between the lattice of ideals of the ring $C(X)$ and the lattice of ideals of the semiring $C^{+}(X)$ are considered. New characterizations of $F$-spaces are obtained. Congruences on abstract semirings are studied. Maximal congruences of semirings $C^+(X)$ are described. It is shown that all congruences on a semifield $U(X)$ of all continuous pozitive functions on $X$ are ideal congruences if and only if $X$ is the pseudocompact space.

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Received: 01.05.1996

Citation: V. I. Varankina, E. M. Vechtomov, I. A. Semenova, “Semirings of continuous nonnegative functions: divisibility, ideals, congruences”, Fundam. Prikl. Mat., 4:2 (1998), 493–510

Citation in format AMSBIB
\by V.~I.~Varankina, E.~M.~Vechtomov, I.~A.~Semenova
\paper Semirings of continuous nonnegative functions: divisibility, ideals, congruences
\jour Fundam. Prikl. Mat.
\yr 1998
\vol 4
\issue 2
\pages 493--510

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    This publication is cited in the following articles:
    1. E. M. Vechtomov, D. V. Chuprakov, “The principal kernels of semifields of continuous positive functions”, J. Math. Sci., 163:5 (2009), 500–514  mathnet  crossref  mathscinet  elib  elib
    2. E. M. Vechtomov, A. V. Cheraneva, “Semifields and their properties”, J. Math. Sci., 163:6 (2009), 625–661  mathnet  crossref  mathscinet  elib  elib
    3. E. M. Vechtomov, D. V. Chuprakov, “Extension of Congruences on Semirings of Continuous Functions”, Math. Notes, 85:6 (2009), 767–779  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. E. M. Vechtomov, A. V. Cheraneva, “Polutela s obrazuyuschei”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 3, 25–33  mathnet  elib
    5. D. V. Chuprakov, “Usloviya distributivnosti reshetki kongruentsii polukolets nepreryvnykh funktsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 3, 128–134  mathnet
    6. E. M. Vechtomov, V. V. Sidorov, “Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions”, J. Math. Sci., 177:6 (2011), 817–846  mathnet  crossref  mathscinet
    7. E. M. Vechtomov, E. N. Lubyagina, “O prostykh idealakh polukolets nepreryvnykh funktsii so znacheniyami v edinichnom otrezke”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 2, 12–18  mathnet
    8. Vechtomov E.M., Lubyagina E.N., “Reshetki nepreryvnykh funktsii so znacheniyami v edinichnom otrezke”, Vestnik Syktyvkarskogo gosudarstvennogo universiteta. Seriya 1: Matematika. Mekhanika. Informatika, 2011, no. 14, 3–20  elib
    9. Shalaginova N.V., “O polukoltsakh rostkov nepreryvnykh neotritsatelnykh funktsii”, Nauchno-tekhnicheskii vestnik Povolzhya, 2011, no. 3, 40–43  elib
    10. E. M. Vechtomov, E. N. Lubyagina, “The determinability of compacts by lattices of ideals and congruencies of semirings of continuous $[0,1]$-valued functions on them”, Russian Math. (Iz. VUZ), 56:1 (2012), 79–82  mathnet  crossref  mathscinet
    11. V. V. Chermnykh, “Functional representations of semirings”, J. Math. Sci., 187:2 (2012), 187–267  mathnet  crossref
    12. E. M. Vechtomov, E. N. Lubyagina, “The semiring of continous $[0,1]$-valued functions”, J. Math. Sci., 191:5 (2013), 633–653  mathnet  crossref
    13. Vechtomov E.M., Lubyagina E.N., “O polukoltsakh sc-funktsii”, Vestnik syktyvkarskogo universiteta. seriya 1: matematika. mekhanika. informatika, 2012, no. 15, 73–82  elib
    14. E. M. Vechtomov, E. N. Lubyagina, “Zamknutye idealy i zamknutye kongruentsii polukolets nepreryvnykh $[0,1]$-znachnykh funktsii s topologiei potochechnoi skhodimosti”, Tr. IMM UrO RAN, 19, no. 3, 2013, 83–93  mathnet  mathscinet  elib
    15. E. M. Vechtomov, V. V. Sidorov, “Opredelyaemost khyuittovskikh prostranstv reshetkami podalgebr polupolei nepreryvnykh polozhitelnykh funktsii s max-slozheniem”, Tr. IMM UrO RAN, 21, no. 3, 2015, 78–88  mathnet  mathscinet  elib
    16. E. M. Vechtomov, N. V. Shalaginova, “Semirings of continuous $(0,\infty]$-valued functions”, J. Math. Sci., 233:1 (2018), 28–41  mathnet  crossref  elib
    17. V. V. Sidorov, “Definability of semifields of continuous positive functions by the lattices of their subalgebras”, Sb. Math., 207:9 (2016), 1267–1286  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    18. E. M. Vechtomov, A. V. Mikhalev, V. V. Sidorov, “Semirings of continuous functions”, J. Math. Sci., 237:2 (2019), 191–244  mathnet  crossref
    19. Sidorov V.V., “Determinability of Semirings of Continuous Nonnegative Functions With Max-Plus By the Lattices of Their Subalgebras”, Lobachevskii J. Math., 40:1, SI (2019), 90–100  crossref  mathscinet  zmath  isi  scopus
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