Subset Product/Examples/Example 4

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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 $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

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 Power Structure, $\set 0$ 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:

\(\ds X \circ_\PP Y\) \(=\) \(\ds \set {x \circ y: x \in X, y \in Y}\) Definition of Operation Induced on Power Set
\(\ds \) \(=\) \(\ds \set 0\) because $\forall x \in X, y \in Y: x \circ y = 0$ by the nature of $\circ$

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 $\struct {\powerset S, \circ_\PP}$.

$\blacksquare$


Sources

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