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What is the relation between semirings in measure theory and semirings in abstract algebra?
Why are they called the same?

You can see : http://en.wikipedia.org/wiki/Semiring

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    There seems to be a typo in your title. Remember: the title is the very first thing people see of your question!2012-09-11

2 Answers 2

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They are different. The name for the object in measure theory is inspired by the object in algebra, viewing $\cap$ as multiplication, $\cup$ as addition, $\emptyset$ as $0$ and $1$ is the union of all sets in $S$ (if this is an element of $S$ - the definition doesn't require that it is).

The difference is that it isn't necessarily closed under those operations, instead we have the condition that, if $A\in S$ and $B\in S$ then there exists a finite number of mutually disjoint sets $C_i \in S$ such that $A\setminus B = \cup_{i=1}^n C_i$.

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    Sure, that's right and that's why there can't be a direct relationship (as one can see by considering the specific example of half-open intervals $[a,b)$ in $\mathbb{R}$). I was not objecting to your conclusion (as I tried to indicate in my last sentence). I was only mildly objecting to viewing $\cup$ as addition.2012-09-11
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The name in algebra comes from the fact that a semiring is a weakened form of a ring in which addition is an abelian semigroup rather than an abelian group.

The name in measure theory comes from looking at a generalization of a ring of sets.