Subset Product/Examples/Example 4

Example of Subset Product
Let $S$ be the initial segment of the natural numbers $\N_{<3}$:
 * $\N_{<3} = \set {0, 1, 2}$

Let $\circ$ be the operation defined on $S$ by the Cayley table:


 * $\begin {array} {c|cccc}

\circ & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ \end {array}$

Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.

Then every non-empty subset of $S$ which does not contain $0$ is invertible in $\struct {\powerset S, \circ_\PP}$.

Proof
By inspection of the Cayley table for $\circ$, it is apparent that $0$ is an identity element of $\struct {S, \circ}$.

From Identity Element for Operation Induced on Power Set, $\set 0$ is an identity element of $\struct {\powerset S, \circ_\PP}$.

Let $X \in \powerset S$ such that $0 \notin X$.

Let $Y \in \powerset S$ be an inverse of $X$ such that $0 \notin Y$.

Then we have:

That is, for this particular algebraic structure, every non-empty subset of $S$ that does not contain $0$ is an inverse of every other non-empty subset of $S$ that does not contain $0$.

It follows that every non-empty subset of $S$ that does not contain $0$ is invertible in $\struct {\powerset S, \circ_\PP}$.