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
REFERENCES
1. Vechtomov E.M. On semirings of partial functions with extended addition. In: International conference dedicated to the 90th anniversary of the Department of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Abstracts of reports, Moscow: Moscow State University Publ., 2019, pp. 20–22 (in Russian).
2. Vechtomov E.M., Lubyagina E.N., Sidorov V.V., Chuprakov D.V. Elementy funktsional’noi algebry [Elements of functional algebra], vol. 1, 384 p, vol. 2, 316 p. Kirov: Raduga-PRESS Publishing House Publ., 2016. ISBN: 975-5-9909330-0-2 .
3. Gelfand I.M., Kolmogorov A.N. On rings of continuous functions on topological spaces. Reports of the Academy of Sciences of the USSR, 1939, vol. 22, no. 1, pp. 11–15 (in Russian).
4. Gratzer G. General lattice theory. Acad. Press, 1978, 380 p. ISBN: 008087391X . Translated to Russian under the title Gratzer G. Obshchaya teoriya reshetok, Moscow: Mir Publ., 1982, 456 p.
5. Sikorski R. Boolean algebras, Berlin, Heidelberg: Springer, 1969, 240 p. ISBN: 9783540044697 . Translated to Russian under the title Sikorski R. Bulevy algebry, Moscow: Mir Publ., 1969, 376 p.
6. Engelking R. General topology. Warsaw, PWN/Polish Scientific Publishers, 1977. ISBN: 0800202090. Translated to Russian under the title R. Engelking, Obshchaya topologiya, Moscow: Mir Publ., 1986, 752 p.
7. Chermnykh V.V. Functional representations of semirings. J. Math. Sci. (NY), 2012, vol. 187, no. 2, pp. 187–267.
8. Gillman L., Henriksen M., Jerison M On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. Proc. Amer. Math. Soc., 1954, vol. 5, no. 3, pp. 447–455.
9. Gillman L., Jerison M. Rings of continuous functions. NY: Springer-Verlag, 1960, 300 p. doi: 10.1007/978-1-4615-7819-2
10. Golan J.S. Semirings and their applications. Dordrecht: Kluwer Acad. Publ., 1999, 382 p. doi: 10.1007/978-94-015-9333-5
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.
