Pairwise Disjoint Sets Partition Any Sets in Semiring

Lemma
Let $$\mathbb S$$ be a semiring of sets.

Let $$A_1, A_2, \ldots, A_n$$ all belong to $$\mathbb S$$.

Then there exists a finite system of pairwise disjoint sets:
 * $$B_1, B_2, \ldots, B_t \in \mathbb S$$

such that every $$A_k$$ where $$k \in \left[{1 \,. \, . \, n}\right]$$ has a finite expansion:
 * $$A_k = \bigcap_{s \in M_k} B_s$$

where $$M_k \subseteq \left[{1 \,. \, . \, s}\right]$$.

That is, the nature of a semiring of sets $$\mathbb S$$ is such that for every collection $$\mathbb T$$ of elements of $$\mathbb S$$, there exists a finite collection of disjoint sets of $$\mathbb S$$ from which you can pick sets from that will build any of the elements of $$\mathbb T$$.

Proof
If $$n = 1$$, the lemma follows trivially from the definition of semiring of sets.

Now we suppose that the lemma holds for $$n = m$$, and we attempt to show it consequently holds for $$n = m+1$$.

We consider a system of sets $$A_1, A_2, \ldots, A_m, A_{m+1}$$ in $$\mathbb S$$.

Let $$B_1, B_2, \ldots, B_t \in \mathbb S$$ satisfy the terms of the lemma for $$A_1, A_2, \ldots, A_m$$.

Now let $$B_{s1} = A_{m+1} \cap B_s$$.

By Pairwise Disjoint Subsets in Semiring Part of Partition, there exists an expansion:
 * $$A_{m+1} = \left({\bigcup_{s=1}^t B_{s1}}\right) \cup \left({\bigcup_{p=1}^q B_{p}'}\right)$$ where $$B_{p}' \in \mathbb S$$.

By definition of a semiring of sets, there is an expansion:
 * $$B_s = B_{s1} \cup B_{s2} \cup \cdots \cup B_{sr_s}$$ where $$B_{sj} \in \mathbb S$$.

It can be seen that:
 * $$A_k = \bigcup_{s \in M_k} \left({\bigcup_{j=1}^{r_s}B_{sj}}\right)$$ for $$k \in \left[{1 \, . \, . \, m}\right]$$

for some $$M_k$$.

Also, the sets $$B_{sj}, B_{p}'$$ are pairwise disjoint.

So, the sets $$B_{sj}, B_{p}'$$ satisfy the conditions of the lemma with respect to the sets $$A_1, A_2, \ldots, A_m, A_{m+1}$$.

Hence the result by induction.