Semirings of continuous partial numerical functions with extended addition

The article deals with the semiring of all continuous functions on a topological space $X$ with values in the topological field of real numbers $\mathbb{R}\cup\{\varnothing\}$, which is completed by the isolated zero $\varnothing$. Operations of addition and multiplication over functions are pointwise. This semiring coincides with the semiring $CP(X)$ of all continuous partial real-valued functions whose domains are clopen subsets of the topological space $X$. The maximal ideals and maximal congruences of the semirings $CP(X)$ are described. A class of maximal subalgebras in the semirings $CP(X)$ is found. It is proved that any Hewitt space $X$ is defined by the semiring $CP(X)$. The case of a finite discrete space $X$ is studied.