26
votes
Accepted
Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?
Following François's suggestion, I ran alg to find a unital commutative semiring which fails to satisfy
$$
\forall x\, y\, z,\; x + z = y + z \land x \times z = y \times z \Rightarrow x = y.
\tag{1}
...
- 48.8k
22
votes
Subtraction-free identities that hold for rings but not for semirings?
The answer to your first question is yes (which was very surprising to me, to be honest). I have no idea whether the second question also has a positive answer. (By the way, don't let the work below ...
- 18.7k
18
votes
Accepted
Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure
There is an extension $R$: take the closure of $\mathbb N$ by the operations $\text{L}$ (or $\varphi$ in the OP) and its inverse $\text{E}$, which are the logarithm and exponential in base $1.2$. ...
- 5,103
12
votes
Categorification of the integers
It should definitely be mentioned here that one often-used categorification of the integers is the sphere spectrum,
$\mathbb S$, i.e. the infinite loop space $\mathbb S = \varinjlim \Omega^n S^n$. I'...
- 63.9k
11
votes
Accepted
Subtraction-free identities that hold for rings but not for semirings?
Tim Campion's idea works, though his example needs a little fixing. As in Tim's answer, we will find a rig with two elements $X$ and $Y$ such that $X+Y=1$ but $XY \neq YX$.
Let $(M,+,0)$ be any ...
- 156k
9
votes
Subtraction-free identities that hold for rings but not for semirings?
The answer to the second question is no in general.
For instance, in an associative ring, the elements $x(x+y)^{-1}$ and $y(x+y)^{-1}$ necessarily commute — in other words, if $a + b = 1$, then $a$ ...
- 63.9k
8
votes
Accepted
If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital?
The answer is no. Let $S$ be a finite meet semilattice without maximum. For concreteness, take $S$ to be the proper subsets of $\{1,2\}$ under intersection. Let $\mathbb NS$ be the semigroup ...
- 38.6k
5
votes
Accepted
What is the derivative of $1/g$ in a differential semiring?
This is not an answer, but is too long for a comment.
It was already mentioned in the comments under the OP that, if $\partial$ is a derivation on a (commutative or non-commutative) semiring $S$ with $...
- 10.5k
5
votes
Examples of $\mathbb{E}_{k}$-semiring spaces
Here are some interesting examples of symmetric bimonoidal groupoids $\mathcal{C}$. In each case, the resulting classifying space $B\mathcal{C}$ is an $E_\infty$ ring space, whose group completion is ...
- 56.9k
4
votes
Accepted
Examples of $\mathbb{E}_{k}$-semiring spaces
(as a sidenote on terminology, as Jonathan points out in the comments, "spectrum" is not really a good name for your objects precisely because you expect them not to be spectra)
If you look ...
- 15.8k
4
votes
The property of category of Semirings
For most "working mathematicians", it's probably enough to be aware of the following result.
Theorem 0: For any Lawvere theory $T$, let $C^T$ be the category of $T$-algebras in $C$, where $C$ is a ...
- 53.3k
4
votes
Categorification of the integers
I think the answer is given by the free Picard groupoid $\mathbb{S}$ on one object. This is the $1$-truncation of the sphere spectrum, keeping only its first two homotopy groups $\mathbb{Z}$ and $\...
- 11.8k
3
votes
Accepted
Name of an algebraic structure that is an idempotent semiring but does not have right distributivity
I'm pretty sure you are looking for a near-semiring. You could call it an "idempotent near-semiring" using left-right if necessary.
There might be other/more terms suggested in Gondran and Minoux's ...
- 1,507
2
votes
Can we have "tropical polynomials" with arbitrary real powers?
I think it will contradict with the definition of a tropical root of a tropical polynomial with multiplicity k, because it is defined as:
Definition 2.4.4. Given a polynomial p(x) with coefficients in ...
2
votes
How do we prove that the following implication in semiring?
Suppose that $d\in \overline{D}$ and $b\in B$ are such that $db\in C^{-1}$; then there exists $c\in C$ such that $db=c^{-1}$. Right-multiplying by $c$ you get $d(bc)=e$. But $bc\in BC\subseteq D$, and ...
- 7,207
2
votes
How to prove the following equivalent condition in idempotent semiring?
The statement is not true. For a counterexample, let $X$ be a set with at least two elements, let $S={\mathcal P}(X)$ be the power set of $X$, let addition in $S$ equal the union operation, and let ...
- 14.6k
2
votes
Accepted
Linear algebra over non-commutative semirings
What I said was mostly right; except that there's no need for the bimodule structure on the $R^n$:
Let $R$ be a semiring and, for $n,m\ge 0$, $R^n$ and $R^m$ the free bimodules over $R$. Say that a ...
- 423
1
vote
Semiring axioms which almost implement inverse, searching for domains other than lambda calculus
This is a sort of strained example but there is certain resemblance to differential operators:
Let $i=0\dots p-2$ and $\mathbb F_p[x_0,\dots x_{p-2}]$ the algebra of polynomials over a finite field.
...
- 2,934
1
vote
Reference request: a cousin to the log semiring
This is not an answer to the OP, but is too long for a comment.
You may want to check Example (1.23) in J.S. Golan's Semirings and their Applications (Springer, 1999). The first sentence reads as ...
- 10.5k
1
vote
Accepted
Define a homomorphism of a set of graphs to its power set
A possible answer is the following, if you are willing to relax the definitions (in a very minor way) of union and intersection of two graphs.
For clarity, let me introduce an additional notation. ...
- 126
1
vote
Are rings really more fundamental objects than semi-rings?
This is not really an answer to the question, but you can really look at both of them together. Let $R$ be any (unital) ring, and let K$_0 (R)$ be the usual Grothendieck group (group generated by ...
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