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Questions tagged [semirings]

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0
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1answer
125 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 ...
1
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0answers
106 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-...
2
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0answers
119 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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1answer
69 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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1answer
643 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-...
0
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1answer
66 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 ...
2
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1answer
53 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 ...
1
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1answer
79 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 ...
4
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3answers
351 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 ...
2
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1answer
164 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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1answer
151 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 ...
3
votes
1answer
221 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 ...
3
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0answers
144 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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1answer
219 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 $p \geq q \geq 0$)? ...
21
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1answer
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 ...
2
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1answer
163 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 ...
7
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0answers
178 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 ...
-1
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1answer
217 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 ...
7
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2answers
397 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 ...
3
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2answers
762 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 ...
3
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0answers
146 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 ...
2
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1answer
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_{-\...
0
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2answers
369 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 ...
2
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1answer
326 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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1answer
332 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) =...
1
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0answers
141 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 $subst(r,s)$ ...
34
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0answers
1k views

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 ...
18
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6answers
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 ...