Ring of Sets Generated by Semiring

Theorem
Let $\mathcal S$ be a semiring of sets.

Let $\mathcal R \left({\mathcal S}\right)$ be the minimal ring generated by $\mathcal S$.

Let $\mathcal L$ be the system of sets $A$ with the finite expansions:
 * $\displaystyle A = \bigcup_{k=1}^n A_k$

with respect to the sets $A_k \in \mathcal S$.

Then $\mathcal L = \mathcal R \left({\mathcal S}\right)$.

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

Let $A, B \in \mathcal L$.

Then by definition of $\mathcal L$, they have expansions:

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

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 $A \cap B \in \mathcal L$ and $A \ast B \in \mathcal L$.

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

From the details of the above construction, the fact that $\mathcal L$ is the minimal ring generated by $\mathcal S$ follows immediately.