2018年10月21日 · I'm reading Halmos's Measure Theory and his definition of semiring seems to disagree with the ones that I find on the internet. Halmos's definition (p. 22): A semiring is a non empty class $\mathbf{P}$ of sets such that

2018年4月28日 · An algebraic structure equipped two binary operations is called a semiring if both the binary operations are associative and one of the operations distributes over the other. Hemiring also seems to be similarly defined.

2019年8月13日 · Shouldn't a regular semiring necessarily satisfy the regularity w.r.t both the operations? Yes, $\mathbb Z$ is a regular semiring in the sense of second definition. But, how do we show its regularity in terms of the first definition? $\endgroup$

If a set function on a semiring is finitely additive and continuous at the empty set, then is it necessarily a premeasure (i.e., countably additive)? 1 relative sequential compactness in L1 along with convergence in measure implies convergence in L1

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.

2016年7月21日 · Yes, that's acceptable. I would say: let $\Sigma$ be a σ-algebra. Then $\Sigma$ satisfies the first two semiring properties because, respectively, $\Sigma$ contains the empty set and $\Sigma$ is closed under finite intersections by virtue of being a σ-algebra.

2016年8月27日 · The 1977 paper by Weiner and Griepentrog uses "semiring" to mean a "left and right distributive seminearring", where what they call a seminearring is almost like Wikipedia's near-semirings, but addition is a semigroup rather than a monoid (so the axiom about absorptivity of 0 is also not imposed). So their "semirings" are like what I asked ...

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

The family of all intervals in the interval $[a,b]$ gives an example of a semiring that is not a ring. I am sure this is an easy task to do; I just want to check if I am correct or not.

Since the intersection of semirings is not necessarily semiring (see is-the-intersection-of-an-arbitrary-collection-of-semirings-a-semiring), the usual method that works for rings (etc.) in this case not works. So (a) it can be proved by a different method (b) there is a counterexample.