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270 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### moduli in real/semi algebraic 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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### 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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### 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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### 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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### 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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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 $(+,\otimes_{-\... 2answers 333 views ### commutative rigs and the Grothendieck Group If I start with a commutative rig, and apply the Grothendieck Group construction to it, twice, once to the additive structure and once to the multiplicative structure, is the result well-known? Does ... 1answer 291 views ### Semirings with subtractive primes Let$S$be a commutative semiring with identity such that each prime ideal of$S$is subtractive. Does this imply all ideals of$S$to be subtractive? By a commutative semiring with identity I mean ... 1answer 298 views ### Dual of idempotent semirings By an idempotent semiring I mean a set equipped with a join-semilattice with bottom structure$(0,+)$and a multiplicative monoid$(1,\cdot)$such that the following equations hold:$a \cdot (b + c) =...
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 $subst(r,s)$ ...