Subalgebras in semirings of continuous partial real-valued functions

This paper refers to the theory of semirings of continuous numerical functions, which has been developed within functional algebra. The object of the investigation is semirings $CP(X)$ of continuous partial functions on topological spaces $X$ with the values in the topological field $\mathbf{R}$ of real numbers. The subject of study is the subalgebras of semirings $CP(X)$. Some properties of the lattices $A(X)$ of all possible subalgebras and $A_1(X)$ of all subalgebras with identity are considered. The structure of atoms and preatoms in lattices $A_1(X)$ is clarified. This allowed us to solve the problem of the absolute determinability of $T_1$-spaces $X$ by each of the lattices $A_1(X)$ and $A_1(X)$.