Semiring

Definition

A semiring $\mathcal{S}$ is a Family of Sets satisfying the following properties

• $\varnothing \in \mathcal{S}$
• $\forall A,B\in \mathcal{S},A\cap B\in \mathcal{S}$
• $\forall A,B\in \mathcal{S},\exists \{B_i{\}}_{i=1}^n\subset \mathcal{S}\text{s.t.}A-B=\underset{i=1}{\overset{n}{⨄}}{B}_i$

where $\uplus$ is a union of the pairwise disjoint sets

Examples

The all below are the semiring on $\Omega =\mathbb{R}$

• $\mathcal{A}=\{(a,b]:-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{[a,b):-\infty <a<b<\infty \}\cup \{\varnothing \}$