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7
votes
2answers
267 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 ...
0
votes
1answer
150 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 ...
22
votes
0answers
893 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 ...
9
votes
0answers
331 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 ...
4
votes
0answers
88 views

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 ...
3
votes
0answers
124 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
votes
0answers
76 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 ...
1
vote
0answers
78 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 ...
1
vote
0answers
121 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 $subst(r,s)$ ...
0
votes
0answers
61 views

What is a necessary and sufficient condition that the kernel of a semi-module homomorphism is a partitioning sub-semi-module?

I would like to identify a representation of the subcategory of a comma category of semi-rings, whose objects are abelian group objects. When attempting to identify the representation, the following ...