Semi-Algebra

Definition

A semi-algebra $\mathcal{A}$ is a Family of Sets satisfying the following properties

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

where $\Omega$ is a universal set