# Definition:Semiring of Sets

## Definition

### Definition 1

Let $\SS$ be a system of sets.

$\SS$ is a semiring of sets or semi-ring of sets if and only if $\SS$ satisfies the semiring of sets axioms:

 $(1)$ $:$ $\ds \O \in \SS$ $(2)$ $:$ $\cap$-stable $\ds \forall A, B \in \SS:$ $\ds A \cap B \in \SS$ $(3)$ $:$ $\ds \forall A, A_1 \in \SS : A_1 \subseteq A:$ $\exists n \in \N$ and pairwise disjoint sets $A_2, A_3, \ldots, A_n \in \SS : \ds A = \bigcup_{k \mathop = 1}^n A_k$

### Definition 2

Let $\SS$ be a system of sets.

$\SS$ is a semiring of sets or semi-ring of sets if and only if $\SS$ satisfies the semiring of sets axioms:

 $(1)$ $:$ $\ds \O \in \SS$ $(2)$ $:$ $\cap$-stable $\ds \forall A, B \in \SS:$ $\ds A \cap B \in \SS$ $(3')$ $:$ $\ds \forall A, B \in \SS:$ $\exists n \in \N$ and pairwise disjoint sets $A_1, A_2, A_3, \ldots, A_n \in \SS : \ds A \setminus B = \bigcup_{k \mathop = 1}^n A_k$

## Also defined as

Some sources specify that a semiring of sets has to be non-empty, but as one of the conditions is that it already contains $\O$, this criterion is superfluous.

## Also see

• Results about semirings of sets can be found here.