\begin{definition}[Definition:Semiring of Sets]
A '''semiring of sets''' or '''semi-ring of sets''' is a system of sets $\SS$, subject to:
:$(1):\quad \O \in \SS$
:$(2):\quad A, B \in \SS \implies A \cap B \in \SS$; 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:
::$(3a):\quad \ds A = \bigcup_{k \mathop = 1}^n A_k$
::$(3b):\quad$ The $A_k$ are pairwise disjoint
Alternatively, criterion $(3)$ can be replaced by:
:$(3'):\quad$ If $A, B \in \SS$, then there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\ds A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.
Some sources specify that a '''semiring of sets''' has to be non-empty, but as one of the conditions is that $\O \in \SS$, this criterion is superfluous.
\end{definition}