E.M. Vechtomov, E.N. Lubyagina. Semirings of continuous partial numerical functions with extended addition ... P. 56-66

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.

Keywords: extended field of real numbers, topological space, semiring of continuous functions, partial function, ideal, congruence, subalgebra, definability

Received October 12, 2022

Revised November 16, 2022

Accepted November 21, 2022

Funding Agency: This work was supported by the Ministry of Science and Higher Education of the Russian Federation under the state contract “Semirings and Their Connections” (project no. 1.5879.2017/8.9).

Evgenii Mikhailovich Vechtomov, Dr. Phys.-Math. Sci., Prof., Vyatka State University, Kirov, 610000, Russia, e-mail: vecht@mail.ru

Elena Nikolaevna Lubyagina, Cand. Sci. (Phys.-Math.), Vyatka State University, Kirov, 610000, Russia, e-mail: shishkina.en@mail.ru

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Cite this article as: E.M. Vechtomov, E.N. Lubyagina. Semirings of continuous partial numerical functions with extended addition. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 56–66.