Definition (*)

A semialgebra (or semiring of sets) $\mathscr{A}\subset2^{X}$ is a collection of sets such that

1. $\emptyset\in\mathscr{A}$
2. $A,B\in\mathscr{A}\implies A\cap B\in\mathscr{A}$
3. For $B\in\mathscr{A}$, there are pairwise disjoint sets $S_{1},\dots,S_{n}\in\mathscr{A}$ s.t. $$\bigsqcup_{j=1}^{n}S_{j}=X\setminus B=B^{c}$$