# Questions tagged [semirings]

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31
questions

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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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184 views

### 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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**1**answer

84 views

### Define a homomorphism of a set of graphs to its power set

Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is,
$G_1\cup G_2$
$=\langle V(G_1)\cup V(G_2), (E(G_1)\...

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194 views

### The property of category of Semirings

I’m now thinking about the property of category of semirings Rig.
Is it complete or co-complete?
I think that Rig has projective and inductive limits, and finite products and co-products, so it ...

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109 views

### On the complexity of writing down matrices

Consider families of $0/1$ matrices in $\Bbb B$ where $1+1=1$:
$\mathcal M_{1,n,c}$ contains $2^n\times 2^n$ matrices that can be written as Hadamard product of $t=O(2^{(\log n)^c})$ matrices $$(J_n-...

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**1**answer

210 views

### Can we have “tropical polynomials” with arbitrary real powers?

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here the notion of a ...

**3**

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**1**answer

84 views

### Name of an algebraic structure that is an idempotent semiring but does not have right distributivity

As the title implies, I am looking for the right name for an algebraic structure, which is exactly as an idempotent semiring, apart from the fact that multiplication does not right-distribute over ...

**20**

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683 views

### Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure

Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...

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70 views

### How do we prove that the following implication in semiring? [closed]

Let $G$ be a group. Clearly the power set $(\mathcal{P}(G),\cup,. )$ is the semiring, where $\cup$ means ordinary union and '.' is defined as $$AB= \left\lbrace ab \in G \mid a\in A\mbox{ and } b\in B ...

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61 views

### How to prove the following equivalent condition in idempotent semiring?

Let $(S,+,.)$ be an idempotent $( a+a=a ~ \forall ~a~ \in S)$ semiring. A partial order on $S$ defined as $a\leq b$ iff $a+b=b$ $\forall ~ a,b \in S$. Note that by an involution function on $S$, we ...

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80 views

### How to prove left linear is left monotone? [closed]

In the paper D. Wilding, M. Johnson, M. Kambites, Exact rings and semirings, J.Algebra 388, (2013), 324-337; doi: j.jalgebra.2013.05.005, arXiv:1212.5358] I found the following statement in page ...

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**3**answers

365 views

### ($\oplus$, $\otimes$) is a semiring. If $\otimes$ = +, what are the possible operators $\oplus$?

Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals.
If $\otimes$ is +, what are the possible operators for $\oplus$?
So far I have proven that ...

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**1**answer

169 views

### Is there a bijection $f: N \times N \rightarrow U \subset N$ with $f(x,y)+f(u,v)=f(x+u,y+v)$ and $f(x,y) \cdot f(u,v)=f(x \cdot u, y \cdot v)$?

Is there a subset of natural numbers that has the same additive and multiplicative structure as the set of ordered pairs of natural numbers under the classical operations of addition and ...

**3**

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156 views

### What is known about semigroups that are generated by (cyclic) subgroups?

A semigroup $(A,\cdot)$ that is idempotent (i.e. $a^2=a$ for every element $a\in A$) is naturally generated by its subgroups (every element on itself constitutes a trivial group). I would like to know ...

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224 views

### Looking for interesting, natural models of this algebraic theory in which $x^\dagger$ is not always the multiplicative inverse of $x$

It is easy to think up interesting, natural models of the algebraic theory presented as follows, such that in these models, $x^\dagger$ is always the multiplicative inverse of $x$.
Question. What ...

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151 views

### Quotients of the initial semiring

The natural numbers are the initial commutative semiring. Thus, for any commutative semiring $R$, there is a unique semiring map $\mathbb{N}\to R$.
For which $R$ is this map an epimorphism?
Some ...

**7**

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**1**answer

226 views

### Semiring of vector bundles on $\mathbb{C}\mathbb{P}^1$

Consider the semiring
$$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$
Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)?
...

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1k views

### Categorification of the integers

I would like to know a natural categorification of the rig of integers $\mathbb{Z}$. This should be a $2$-rig. Among the various notions of 2-rigs, we obviously have to exclude those where $+$ is a ...

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**1**answer

169 views

### Terminology for the equation $a=a+b$ in commutative semigroups

Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often ...

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184 views

### Moduli in semialgebraic geometry

Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry?
The sort of thing I am imagining is an object in a category of semischemes:
Ordinary schemes ...

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**1**answer

221 views

### An extension of the real semiring with multiple degrees of infinity

Is it possible to define an extension of the probability semiring $(\mathbb{R}^+, +, \times, 0, 1)$ such that
Closure $a^* = 1 + a + a^2 + \ldots$ is defined for every element of the semiring, not ...

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**2**answers

411 views

### Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...

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**2**answers

840 views

### Semiring naturally associated to any monoid?

For any monoid $M$, we can naturally construct a semiring $S$ as follows:
Let the additive monoid of $S$ be the free commutative monoid on $M$
Let the multiplicative monoid of $S$ be $M$
Then, if ...

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151 views

### Semirings where solving linear systems is in P

Solving linear systems appears hard in semirings.
In $\mathbb{N}_0 (+,\times)$ it is NP-complete via reduction to 1-in-3 SAT.
In the min-plus semiring the complexity is $ NP \cap coNP$ according to ...

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**1**answer

137 views

### q-product semiring

q-product is defined as
$x \otimes _q y = (x^{1-q}+y^{1-q}-1)^{1/(1-q)}$
Observation:
$(+,\otimes_\infty)$ is min-plus tropical semiring on the segment $[0,1]$
$(+,\otimes_1)$ is R
$(+,\otimes_{-\...

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403 views

### 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 ...

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345 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 ...

**3**

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**1**answer

344 views

### Dual of idempotent semirings

By an idempotent semiring I mean a set equipped with a join-semilattice with bottom structure $(0,+)$ and a multiplicative monoid $(1,\cdot)$ such that the following equations hold:
$a \cdot (b + c) =...

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148 views

### Substitution semiring?

Let G be a [ CF ] grammar, and let elements of semiring be sets of rules.
Define multiplication as:
$$ x\otimes y = \{ t| \exists r \in x \exists s \in y (t=subst(r,s))\} $$
where $subst(r,s)$ ...

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### Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...

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2k views

### Are rings really more fundamental objects than semi-rings?

The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics.
From then on, it seems ...