Axiom:Semiring of Sets Axioms/Axioms 1

Definition
Let $\SS$ be a system of sets.

$\SS$ satisfies the semi-ring of sets axioms $\SS$ satisfies the following axioms:
 * $(1):\quad \O \in \SS$
 * $(2):\quad A, B \in \SS \implies A \cap B \in \SS$; that is, $\SS$ is $\cap$-stable
 * $(3):\quad A, A_1 \in \SS : A_1 \subseteq A \implies \exists$ a finite sequence of pairwise disjoint sets $A_2, A_3, \ldots, A_n \in \SS :$
 * $\ds A = \bigcup_{k \mathop = 1}^n A_k$

Also see

 * Axiom:Semiring of Sets Axioms/Axioms 2, for an alternative formulation of the axioms of a semiring of sets.


 * Definition:Semiring of Sets


 * Equivalence of Definitions of Semiring of Sets