Definition:Semiring of Sets

A semiring of sets $$\mathcal {S}$$ is a non-empty 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 $$A$$.