Ring of Sets Generated by Semiring

Theorem
Let $\SS$ be a semiring of sets.

Let $\map \RR \SS$ be the minimal ring generated by $\SS$.

Let $\LL$ be the system of sets $A$ with the finite expansions:
 * $\ds A = \bigcup_{k \mathop = 1}^n A_k$

with respect to the sets $A_k \in \SS$.

Then $\LL = \map \RR \SS$.

Proof
First we need to show that $\LL$ is a ring of sets.

Let $A, B \in \LL$.

Then by definition of $\LL$, they have expansions:

Since $\SS$ is a semiring of sets, we have:
 * $C_{ij} = A_i \cap B_j \in \SS$

By Pairwise Disjoint Subsets in Semiring Part of Partition, there exist finite expansions:

From these, it follows that $A \cap B$ and $A \ast B$ have the finite expansions:

Hence both $A \cap B \in \LL$ and $A \ast B \in \LL$.

So by definition, $\LL$ is a ring of sets.

From the details of the above construction, the fact that $\LL$ is the minimal ring generated by $\SS$ follows immediately.