Definition:Semiring of Sets

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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.