Questions tagged [semirings]
The semirings tag has no usage guidance.
45
questions
2
votes
1
answer
144
views
Semiring axioms which almost implement inverse, searching for domains other than lambda calculus
I'm working with an idempotent semiring which contains elements $C_i, \hat{C_i}$ with the following properties:
$$ {C}_i \hat{C_j} = 0 \quad\text{where}\quad i \neq j \quad\quad\quad\quad(\beta_0)$$
$$...
- 225
4
votes
0
answers
146
views
History of tropical mathematics
This is a follow-up to this question about the origin of tropical mathematics.
Are there any articles, websites or books which deal with the history of tropical mathematics?
I have been trying to find ...
- 211
1
vote
0
answers
66
views
When is the preorder on a semi-ring a lattice?
Each semi-ring $R$ comes equipped with a canonical preorder $r\leq r^\prime \Leftrightarrow \exists w: r + w = r^\prime$. If $R$ is a ring this order collapses. However, if $R$ is the positive part of ...
3
votes
1
answer
100
views
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 ...
- 24.9k
4
votes
0
answers
174
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}(\...
- 9,575
3
votes
0
answers
174
views
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 ...
- 9,575
6
votes
0
answers
153
views
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 ...
- 9,575
5
votes
2
answers
423
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})$, ...
- 9,575
26
votes
3
answers
704
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$ ...
- 32.2k
8
votes
1
answer
389
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 ...
- 183
1
vote
1
answer
275
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 ...
- 15.1k
2
votes
0
answers
256
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 ...
2
votes
1
answer
272
views
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 ...
- 12.4k
23
votes
1
answer
868
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 ...
- 14.7k
0
votes
0
answers
72
views
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 ...
- 203
6
votes
1
answer
331
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 ...
- 423
2
votes
1
answer
147
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)\...
- 203
1
vote
1
answer
352
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 ...
- 143
1
vote
0
answers
114
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-...
- 13.5k
2
votes
1
answer
260
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 ...
- 2,136
3
votes
1
answer
119
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 ...
- 151
21
votes
1
answer
737
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-...
- 11.7k
0
votes
1
answer
74
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
votes
1
answer
73
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
vote
1
answer
81
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 ...
3
votes
3
answers
439
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 ...
- 133
2
votes
1
answer
186
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 ...
- 175
3
votes
1
answer
178
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
1
answer
230
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 ...
- 3,653
3
votes
0
answers
189
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 ...
- 63.6k
7
votes
1
answer
260
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$)?
...
- 59.2k
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 ...
- 59.2k
2
votes
1
answer
176
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
votes
0
answers
204
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 ...
- 101
-1
votes
1
answer
246
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
votes
2
answers
449
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 ...
- 26.8k
3
votes
2
answers
1k
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 ...
- 4,333
3
votes
0
answers
158
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 ...
- 24k
2
votes
1
answer
148
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_{-\...
- 244
0
votes
2
answers
516
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 ...
- 11.6k
2
votes
1
answer
362
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
votes
1
answer
379
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) =...
- 1,261
1
vote
0
answers
169
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 $subst(r,s)$ ...
- 244
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 ...
- 6,079
22
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 ...