Equivalence of Definitions of Semiring of Sets

Theorem
A collection $\SS$ of subsets of a set $X$ is a semiring (of sets) :
 * $(1):\quad \O \in \SS$
 * $(2):\quad A, B \in \SS \implies A \cap B \in \SS$; that is, $\SS$ is $\cap$-stable
 * $(3):\quad$ If $A, A_1 \in \SS$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \SS$ such that:
 * $(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$
 * $(3b):\quad$ The $A_k$ are pairwise disjoint

We prove that criterion $(3)$ can be replaced by:
 * $(3'):\quad$ If $A, B \in \SS$, then there exist finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.

$(3)$ implies $(3')$
Let $X$ be a set, and let $\SS$ be a collection of subsets of $X$.

Suppose that for all $A, A_1 \in \SS$ such that $A_1 \subseteq A$, there exists a finite sequence of sets $A_2, A_3, \ldots, A_n \in \SS$ such that:
 * $A_1, A_2, \ldots, A_n$ are pairwise disjoint
 * $\displaystyle A = \bigcup_{k \mathop = 1}^n A_k$

Let $B \in \SS$, and let $A_1 = A \cap B$.

It follows that $A_1 \in \SS$, by definition.

Also, $A_1 \subseteq A$ by Intersection is Subset.

Then:

as required.

$(3')$ implies $(3)$
Now suppose that for all $A, B \in \SS$, there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.

Then $B$ is disjoint with each of the sets $A_k$.

Let $B \subseteq A$. Then:

as required.