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:
 * $$A = \bigcup_{k=1}^n A_k$$

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.