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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4 votes
0 answers
153 views

Does the tropical semiring admit a universal property?

Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
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What is the initial semiring category with a (commutative) semiring?

Recall that The biinitial monoidal category with a monoid is given by the augmented simplex category together with the monoid $([0],\sigma^{0}_{0},\delta^{0}_{0})$ there. The biinitial symmetric ...
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Has this notion of a ring in a bimonoidal category been studied before?

The Baez–Dolan microcosm principle is stated in the nLab as follows. Microcosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same ...
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5 votes
2 answers
379 views

Examples of $\mathbb{E}_{k}$-semiring spaces

Semirings, also called rigs, are rings without negatives: their underlying additive monoids are not groups (in other words, while rings are monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, ...
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26 votes
3 answers
659 views

Subtraction-free identities that hold for rings but not for semirings?

Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in: Question 1. Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ ...
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8 votes
1 answer
348 views

If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital?

Suppose $S$ is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring $G(S)$ is a unital ring. Can we conclude that $S$ must be ...
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1 answer
176 views

Reference request: a cousin to the log semiring

Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
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2 votes
0 answers
205 views

Union star symbol in set theory

In the slides Provenance for Database Transformations, page 24, they provide a semiring for lineage, which include a $\cup^*$ symbol. However, I can not find any related materials about the meaning of ...
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2 votes
1 answer
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What is the derivative of $1/g$ in a differential semiring?

Let $(S,+,\cdot)$ be a semiring; a derivation on $S$ is a map $\partial : S \to S$ that is linear and Leibniz, in the sense that It is a semigroup homomorphismm with respect to $+$; $\partial(a\cdot ...
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23 votes
1 answer
741 views

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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1 answer
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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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2 votes
1 answer
136 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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1 answer
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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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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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2 votes
1 answer
248 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 ...
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3 votes
1 answer
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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 ...
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21 votes
1 answer
719 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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0 votes
1 answer
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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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2 votes
1 answer
69 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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1 vote
1 answer
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 votes
3 answers
414 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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2 votes
1 answer
177 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 ...
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3 votes
1 answer
163 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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3 votes
1 answer
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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3 votes
0 answers
172 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 ...
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7 votes
1 answer
249 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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27 votes
2 answers
2k 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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2 votes
1 answer
173 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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7 votes
0 answers
196 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 votes
1 answer
237 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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7 votes
2 answers
432 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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3 votes
2 answers
945 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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3 votes
0 answers
155 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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2 votes
1 answer
139 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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0 votes
2 answers
479 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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2 votes
1 answer
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 ...
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3 votes
1 answer
360 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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1 vote
0 answers
159 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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40 votes
0 answers
2k 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 ...
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20 votes
6 answers
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 ...