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Algebra Basics

The only algebra on \(\Omega=\emptyset\) is the collection \(\Omega\).

The only algebra on \(\Omega=\{1\}\) is the collection \(\emptyset,\Omega\).

The only two algebras on \(\Omega=\{1,2\}\) are \(\emptyset,\Omega\) and \(\mathcal P(\Omega)\).

There are exactly five algebras on \(\Omega=\{1,2,3\}\).

1. \(\emptyset,\Omega\)

2. \(\mathcal P(\Omega)\)

3. \(\{\emptyset,\{1\},\{2,3\},\Omega\}\)

4. \(\{\emptyset,\{2\},\{1,3\},\Omega\}\)

5. \(\{\emptyset,\{3\},\{1,2\},\Omega\}\)

Max and Min

We call \(\{\emptyset,\Omega\}\) the minimal algebra on \(\Omega\). We call \(\mathcal P(\Omega)\) the maximal algebra on \(\Omega\). These are both always algebras for any set \(\Omega\).

Proof: \(\mathcal P(\Omega)\) always satisfies every containment requirement because it contains everything. Therefore, trivially, this satisfies the conditions for being an algebra.

To check that \(\{\emptyset,\Omega\}\) is an algebra, trivially it contains the universe. It is closed under complements because \(\emptyset^c=\Omega\in\{\emptyset,\Omega\}\) and \(\Omega^c=\emptyset\in\{\emptyset,\Omega\}\). It is closed under unions because

• \(\emptyset\cup\emptyset=\emptyset \in\{\emptyset,\Omega\}\)

• \(\emptyset\cup\Omega=\Omega\cup\emptyset=\Omega\in\{\emptyset,\Omega\}\)

• \(\Omega\cup\Omega=\Omega\in\{\emptyset,\Omega\}\)