Category:Semirings of Sets

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This category contains results about Semirings of Sets.
Definitions specific to this category can be found in Definitions/Semirings of Sets.

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 $A_2, A_3, \ldots, A_n \in \SS : \ds A = \bigcup_{k \mathop = 1}^n A_k$