# Definition:Semiring of Sets

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## Definition

A **semiring of sets** or **semi-ring of sets** is a system of sets $\mathcal S$, subject to:

- $(1):\quad \varnothing \in \mathcal S$
- $(2):\quad A, B \in \mathcal S \implies A \cap B \in \mathcal S$; i.e., $\mathcal S$ is $\cap$-stable
- $(3):\quad$ If $A, A_1 \in \mathcal S$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \mathcal S$ such that:
- $(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$
- $(3b):\quad$ The $A_k$ are pairwise disjoint

Alternatively, criterion $(3)$ can be replaced by:

- $(3'):\quad$ If $A, B \in \mathcal S$, then there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \mathcal S$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.

Some sources specify that a **semiring of sets** has to be non-empty, but as one of the conditions is that $\varnothing \in \mathcal S$, this criterion is superfluous.

## Also see

- Results about
**semirings of sets**can be found here.

- Set of Empty Set is Semiring of Sets: $\mathcal S = \left\{{\varnothing}\right\}$ is a
**semiring of sets** - Half-Open Rectangles form Semiring of Sets: the half-open $n$-rectangles form a
**semiring of sets** - Cartesian Product of Semirings of Sets: the Cartesian product of two
**semirings of sets**is again a**semiring of sets**

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 6$