Definition:Semiring of Sets

Definition
A semiring of sets $\mathcal S$ is a system of sets such that:


 * $\varnothing \in \mathcal S$;


 * $A, B \in \mathcal S \implies A \cap B \in \mathcal S$;


 * If $A, A_1 \in \mathcal S$ such that $A_1 \subseteq A$, then $\exists A_2, A_3, \ldots, A_n \in \mathcal S$ such that $A$ can be expressed as:
 * $\displaystyle A = \bigcup_{k=1}^n A_k$

where $A_1, A_2, \ldots, A_n$ forms a partition of $A$.

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

Note in passing that, by this definition, $\mathcal S = \left\{{\varnothing}\right\}$ is itself a semiring of sets.