Semirings of continuous $(0,\infty]$-valued functions

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.