1 results
FROM TOPOLOGIES OF A SET TO SUBRINGS OF ITS POWER SET
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 1 / August 2020
- Published online by Cambridge University Press:
- 20 February 2020, pp. 15-20
- Print publication:
- August 2020
-
- Article
- Export citation
-
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 
$n-1$.
