# Questions tagged [semirings]

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### Quiver algebras from semirings and posets as semirings

A semiring is a nonempty set $S$ such two binary operations + and * making S into a semigroup with + and * and such that a*(b+c)=ab+ac and (b+c)a=ba+c*a for all a,b,c in S. Assume in the following ...
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### Partition of multisets of polynomials

Problem: Given a multiset $S$ of irreducible polynomials in $\mathbb{Z}[x]$, say YES if $S$ can be partitioned into two nonempty multisets $A$ and $B$ such that both the product of all the elements of ...
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### Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is ...
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### A semifield of characteristic zero may have a finite number of elements

A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$. I ...
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### Linear algebra over non-commutative semirings

I'm reading up on linear algebra over semirings, and I'm wondering why people seem to stop short of showing an equivalence between linear transformations between free modules and matrices. It seems ...
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### commutative rigs and the Grothendieck Group

If I start with a commutative rig, and apply the Grothendieck Group construction to it, twice, once to the additive structure and once to the multiplicative structure, is the result well-known? Does ...
357 views

### Semirings with subtractive primes

Let $S$ be a commutative semiring with identity such that each prime ideal of $S$ is subtractive. Does this imply all ideals of $S$ to be subtractive? By a commutative semiring with identity I mean ...
360 views