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:
 * $\displaystyle A_k = \bigcap_{s \in M_k} B_s$

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:


 * $\displaystyle A_{m+1} = \left({\bigcup_{s=1}^t B_{s1}}\right) \cup \left({\bigcup_{p=1}^q B_{p}'}\right)$ where $B_{p}' \in \mathbb S$.

By definition of a semiring of sets, there is an expansion:


 * $\displaystyle B_s = B_{s1} \cup B_{s2} \cup \cdots \cup B_{sr_s}$ where $B_{sj} \in \mathbb S$.

It can be seen that:


 * $\displaystyle A_k = \bigcup_{s \in M_k} \left({\bigcup_{j=1}^{r_s}B_{sj}}\right)$ 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.