Definition:Algebra of Sets

Definition 1
Given a set $$X \ $$ and a collection of subsets of $$X \ $$, $$\mathcal{S} \subset \mathcal{P} \left({X}\right) \ $$, $$\mathcal{S} \ $$ is called an algebra of sets if, given that $$A, B \in \mathcal{S} \ $$,


 * 1) $$A \cup B \in \mathcal{S} \ $$
 * 2) $$\mathcal{C}_X \left({A}\right) \in \mathcal{S} \ $$

where $$\mathcal{C}_X \left({A}\right)$$ is the relative complement of $$A \ $$ in $$X$$.

Definition 2
An algebra of sets is a ring of sets with a unit.

The two definitions are equivalent.

Power Set
The Power Set is Algebra of Sets.

Null Set and Set Itself
Let $$S$$ be any non-empty set.

Then $$\left\{{S, \varnothing}\right\}$$ is (trivially) an algebra of sets, where $$S$$ is the unit.

Historical Note
The concept of an algebra of sets was invented by George Boole, after whom Boolean algebra was named.