Algebra generated by a collection of subsets of a set X

Question

We define algebra generated by a subset S of power set of X as intersection of all algebras containing S, Is there a procedure of finding this algebra generated. Just like we find subspace generated by a subset of a vector space as span of that set i.e. taking all linear combinations of elements of that subset.

Answer 1

A good example of an algebra is the sigma algebra of Borel sets. For a description of Borel Sets, see Borel Hierarchy. If you're working in a separable, completely metrizable metric space, (Polish space), then this is the smallest sigma algebra containing the open sets. This hierarchy is stratified, and each level has a description because it is generated by the operations of $\cap$, $\cup$, or complementation. As you know, an algebra is closed under finite intersections, unions, and complementation. Following a similar scheme, you might be able to precisely describe the sets that comprise other algebras.

Answer 2

Suppose $S$ is a finite set, $S = \{S_1, \dots, S_n\}$. Let be $N =\{1,2\dots, n+1\}$, $S_{n+1} = (\cup_i S_i)^c$ and $S' = S\cup \{S_{n+1}\}$. Define sets $$T_I = \left(\bigcup_{j\in I} S_j\right)\backslash \left(\bigcup_{j\notin I} S_j\right)$$ for $I\subseteq N$, i. e. $I\in \mathcal{P}(N)$. Make some Venn-diagram-like drawings to visualize what $T_I$'s are and note that $$\bigcup_\emptyset = \emptyset\text{.}$$

Then, you can see that the algebra generated by $S$ (which equals the algebra qenerated by $S'$) consist precisely of all sets

$$E_J =\bigcup_{I\in J} T_I\text{,}$$ where $J\subseteq\mathcal{P(N)}$. I like to think of $T_I$'s as of "base sets" or as of an analogue of complete system of events in probability theory. This approach, however, doesn't work when the set $S$ is infinite.