Geodesic
Semirings of continuous nonnegative functions: divisibility, ideals, congruences
Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 493-510 
V. I. Varankina; E. M. Vechtomov; I. A. Semenova. Semirings of continuous nonnegative functions: divisibility, ideals, congruences. Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 493-510. http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a1/
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     author = {V. I. Varankina and E. M. Vechtomov and I. A. Semenova},
     title = {Semirings of continuous nonnegative functions: divisibility, ideals, congruences},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {493--510},
     year = {1998},
     volume = {4},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a1/}
}
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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.