Geodesic
Semirings of continuous $(0,\infty]$-valued functions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 6, pp. 43-64.

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The semiring $C^{\infty}(X)$ of all continuous functions on an arbitrary topological space $X$ with values in the topological semiring $(0,\infty]$ is studied. General properties of semirings $C^\infty(X)$ are considered. Properties of lattices of ideals and congruences of semirings $C^{\infty}(X)$ over the $\mathrm{P}$-spaces $X$, the $\mathrm{F}$-spaces $X$, and the finite discrete spaces are proved. 
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E. M. Vechtomov; N. V. Shalaginova. Semirings of continuous $(0,\infty]$-valued functions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 6, pp. 43-64. http://geodesic.mathdoc.fr/item/FPM_2015_20_6_a2/

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