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 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 A1(X) of all subalgebras with identity are considered. The structure of atoms and preatoms in lattices A1(X) is clarified. This allowed us to solve the problem of the absolute determinability of T1-spaces X by each of the lattices A1(X) and A1(X). © 2023, Springer Nature Switzerland AG.