Definition:Semiring of Sets
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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
- Set of Empty Set is Semiring of Sets
- Half-Open Rectangles form Semiring of Sets
- Cartesian Product of Semirings of Sets
- Results about semirings of sets can be found here.
Sources
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- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 6$