Definability of \(T_1\)-spaces by the lattice of subalgebras of semirings of continuous partial real-valued functions on them. (Russian. English summary) Zbl 1403.16041

Vechtomov, E. M.; Lubyagina, E. N.

Vestn. Syktyvkar. Univ., Ser. 1, Mat. Mekh. Inform. 22, No. 1, 21-28 (2017).

Summary: The article refers to the general theory of semirings of continuous functions. We consider subalgebras of semirings \(CP(X)\) of continuous partial functions on topological spaces \(X\) with values in the topological field \(\mathbb R\) of real numbers. We study the minimal and maximal subalgebras of the \(\mathbb R\)-algebra \(CP(X)\). We prove a definability theorem of an arbitrary \(T_1\)-space \(X\) by the lattice \(A(X)\) of all subalgebras of the semiring \(CP(X)\).

MSC:

16Y60	Semirings
03C40	Interpolation, preservation, definability

Keywords:

semiring; field of real numbers; partial real-valued function; subalgebra