Definition:Semiring of Sets

Definition
A semiring of sets or semi-ring of sets is a system of sets $\mathcal S$, subject to:


 * $(1):\quad \varnothing \in \mathcal S$
 * $(2):\quad A, B \in \mathcal S \implies A \cap B \in \mathcal S$; i.e., $\mathcal S$ is $\cap$-stable
 * $(3):\quad$ If $A, A_1 \in \mathcal S$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \mathcal S$ such that:
 * $(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$
 * $(3b):\quad$ The $A_k$ are pairwise disjoint

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

Some sources specify that a semiring of sets has to be non-empty, but as one of the conditions is that $\varnothing \in \mathcal S$, this criterion is superfluous.

The above two definitions are equivalent.

Examples

 * $\mathcal S = \left\{{\varnothing}\right\}$ is a semiring of sets (proof)
 * The half-open $n$-rectangles form a semiring of sets (proof)
 * The Cartesian product of two semirings of sets is again a semiring of sets (proof)