Abstract
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).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Engelking, General Topology [Russian translation], Mir, Moscow (1986).
I. M. Gelfand and A. N. Kolmogorov, “On rings of continuous functions on topological spaces,” Dokl. Akad. Nauk SSSR, 22, No. 1, 11–15 (1939).
L. Gillman and M. Jerison, Rings of Continuous Functions, New York (1976).
J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht (1999).
G. Gretzer, General Theory of Lattices [Russian translation], Mir, Moscow (1982).
E. Hewitt, “Rings of real-valued continuous functions. I,” Trans. Am. Math. Soc., 64, No. 1, 45–99 (1948).
V. V. Sidorov, “Isomorphisms of lattices of subalgebras of semifields of positive continuous functions,” Sib. Math. J., 60, No. 3, 526–541 (2019).
V. V. Sidorov, “Isomorphisms of semirings of continuous nonnegative functions and the lattices of their subalgebras,” Lobachevskii J. Math., 41, No. 9, 1684–1692 (2020).
E. M. Vechtomov and E. N. Lubyagina, “Semirings of continuous partial real-valued functions” [in Russian], in: Proc. of the 48th Int. Youth School-Conference “Modern Problems in Mathematics and Its Applications,” Yekaterinburg, 2017, CEUR-WS.org, Vol. 1894, pp. 20–29.
E. M. Vechtomov, E. N. Lubyagina, V. V. Sidorov, and D. V. Chuprakov, Elements of Functional Algebra [in Russian], Raduga-Press, Kirov (2016).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 24, No. 1, pp. 125–140, 2022.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vechtomov, E.M., Lubyagina, E.N. Subalgebras in Semirings of Continuous Partial Real-Valued Functions. J Math Sci 269, 697–707 (2023). https://doi.org/10.1007/s10958-023-06307-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06307-2