"Ring" and "semiring" are concepts defined both in algebra and set theory.
In Algebra
A ring in algebra is a set R equipped with two binary operations + and · called addition and multiplication, that Addition (+) is abelian, Multiplication (⋅) is associative, Multiplication distributes over addition, and Multiplicative identity (1) exists.
A semiring in abstract algebra, is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
In Set theory
A ring of sets in measure theory is a family of sets closed under unions and set-theoretic differences. That is, it obeys the two properties
$$A \setminus B \in \mathcal{R} $$
$$A \cup B \in \mathcal{R}$$
This implies that it is also closed under intersections,
$$A \cap B \in \mathcal{R}$$
A semiring of sets is a non-empty collection S of sets such that
- $\emptyset \in S$
- If $E \in S$ and $F \in S$ then $E \cap F \in S$.
- If $E \in S$ and $F \in S$ then there exists a finite number of
mutually disjoint sets $C_i \in S$ for $i=1,\ldots,n$ such that $E
\setminus F = \bigcup_{i=1}^n C_i$.
I wonder, are such definitions somehow linked or equivalent? or they are just coincidence that both fields of mathematics used the same terms?
Best Answer
Using your definition of "ring of sets in measure theory", that it is a (nonempty) collection of sets $R$ such that (1) it is closed under union ($\forall A,B\in R$ we have $A\cup B\in R$) and (2) it it is closed under set-theoretic difference ($\forall A,B\in R$ we have $A-B\in R$), we can in fact show $R$ is a commutative ring in the algebra sense, however with respect to the operations symmetric difference and intersection. So first we show that in fact, any ring of sets in measure theory is closed over symmetric difference and intersection:
Lemma. Let $R$ be a ring of set, then for all $A,B\in R$, we have $A\Delta B\in R$ and $A\cap B\in R$. (Here $A\Delta B:=(A-B)\cup(B-A)$ is the operation symmetric difference.)
Pf. The closure of symmetric difference is clear from its definition. And note well that $A\cap B= (A\cup B)-(A\Delta B)$. $\Box$
Now we make the observation: If $R$ is a ring of sets, then $\varnothing\in R$, since for any $A\in R$, we have $A\Delta A=\varnothing \in R$. Further note that for any $A\in R$, we have $\varnothing\Delta A=A$. Hence if we identify the operation $\Delta$ as "addition" and $\varnothing$ as the "additive identity", and every $A\in R$ is its own "additive inverse", we have
Claim. If $R$ is a ring of sets, then $(R,\Delta)$ is an abelian gorup. $\Box$ (Associativity given set-theoretically.)
Now, identify the operation $\cap$ as "multiplication", then with the following
Fact. Intersection distributes over symmetric difference. $\Box$
we have finally:
Proposition. If $R$ is a ring of set (measure theory sense), then it is also closed under symmetric difference $\Delta$ and intersection $\cap$. And that $(R,\Delta,\cap)$ forms a commutative ring (algebra sense). $\Box$