Subset Relation on Power Set is Partial Ordering

Theorem

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the subset relation $\subseteq$.

Then $\struct {\powerset S, \subseteq}$ is an ordered set.

The ordering $\subseteq$ is partial if and only if $S$ is neither empty nor a singleton; otherwise it is total.

From Subset Relation is Ordering, we have that $\subseteq$ is an ordering on any set of subsets of a given set.

Suppose $S$ is neither a singleton nor the empty set.

Then $\exists a, b \in S$ such that $a \ne b$.

Then $\set a \in \powerset S$ and $\set b \in \powerset S$.

However, $\set a \nsubseteq \set b$ and $\set b \nsubseteq \set a$.

So by definition, $\subseteq$ is a partial ordering.

Now suppose $S = \O$.

Then $\powerset S = \set \O$ and, by Empty Set is Subset of All Sets, $\O \subseteq \O$.

Hence, trivially, $\subseteq$ is a total ordering on $\powerset S$.

Now suppose $S$ is a singleton: let $S = \set a$.

Then $\powerset S = \set {\O, \set a}$.

So there are only two elements of $\powerset S$, and we see that $\O \subseteq \set a$ from Empty Set is Subset of All Sets.

So, trivially again, $\subseteq$ is a total ordering on $\powerset S$.

$\blacksquare$

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.2$.Subsets: Example $11$
1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.1$: Partially ordered sets: Example $3$
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$: Example $7.1$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets: $\text{(i)}$
1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $2 \ \text {(a)}$