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FROM TOPOLOGIES OF A SET TO SUBRINGS OF ITS POWER SET

Part of: Combinatorics Algebraic combinatorics Finite commutative rings Algebraic logic Ring extensions and related topics Complex dynamical systems Chain conditions, finiteness conditions

Published online by Cambridge University Press:  20 February 2020

ALI JABALLAH Open the ORCID record for ALI JABALLAH [Opens in a new window]  and
NOÔMEN JARBOUI Open the ORCID record for NOÔMEN JARBOUI [Opens in a new window]
ALI JABALLAH
Affiliation:
Department of Mathematics, University of Sharjah, P.O. Box 27272, Sharjah, UAE email ajaballah@sharjah.ac.ae
NOÔMEN JARBOUI*
Affiliation:
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa31982, Saudi Arabia Département de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, Route de Soukra, P.O. Box 1171, Sfax3038, Tunisia email njarboui@kfu.edu.sa
*
✉njarboui@kfu.edu.sa
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Abstract

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

Keywords

integral extensionsubringintermediate ringBoolean algebrafinite topologypartitionBell numbers

MSC classification

Primary: 05E40: Combinatorial aspects of commutative algebra
Secondary: 03G05: Boolean algebras 05A18: Partitions of sets 06E05: Structure theory 06E10: Chain conditions, complete algebras 13B02: Extension theory 13B21: Integral dependence; going up, going down 13E99: None of the above, but in this section 13M05: Structure 13M99: None of the above, but in this section 37F20: Combinatorics and topology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 15 - 20
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author thanks the University of Sharjah for funding Research Project No. 1902144081.

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