Questions tagged [semirings]
The semirings tag has no summary.
54 questions
8
votes
2
answers
346
views
Let $k$ be a semiring. Show that the semiring of formal power series $k[[x]]$ is not finitely generated as a $k$-algebra
I asked a similar question on
math.stackexchange, which unfortunately did not receive a satisfactory answer..
Let $k$ be a commutative semiring (with at least two elements) and let $k[[x]]$ be the ...
- 933
4
votes
0
answers
107
views
Do all commutative semirings satisfy the Orzech property?
Reutenauer & Straubing (1984) showed that all commutative semirings are stably finite, and Yi-Jia Tan (2016) showed that nontrivial commutative semirings satisfy the strong rank condition (Theorem ...
- 1,354
3
votes
1
answer
236
views
Additively idempotent semirings that are not lattices
I am looking for examples of additively idempotent semirings (which are always join semilattices) that do not have an underlying lattice structure, i.e. either meets do not exist or exist outside the ...
- 31
7
votes
0
answers
193
views
Reference request for an algebraic structure mimicking $\varepsilon_0$
Let $(R, +, 0, \cdot, 1)$ be a (unital) semiring equipped with a unary operation $f$ and a binary operation $\land$ such that
$f(0) = 1$
$f(x + y) = f(x) f(y)$
$x \land y = y \land x$
$x \land (y \...
- 2,055
4
votes
1
answer
293
views
Do all commutative semirings satisfy the strong rank condition?
It is well known that all nonzero commutative rings satisfy the strong rank condition (and in fact also the potentially stronger Orzech property; whether they are equivalent for rings is the subject ...
- 1,354
4
votes
0
answers
105
views
Reference request: right local semirings
Two months ago I asked on Math.SE (1) whether there exists a semiring that has a unique maximal left ideal but multiple maximal right ideals, and (2) whether a semiring could have a unique maximal ...
- 1,354
2
votes
1
answer
1k
views
Are there infinitely many $g$- or $m$-primes?
I will use the definition of $f_n(x)$ from here. Let us call a number $n$ a g-number (g = for "gerade = even" in German) if for all $x\in \mathbb{R}: f_n(x)=f_n(-x)$.
Let us call a number $n$...
13
votes
1
answer
627
views
Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 2,055
2
votes
0
answers
565
views
Are these finite semirings known?
I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...
2
votes
1
answer
232
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
260
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
158
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
177
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 ...
- 28.4k
4
votes
0
answers
258
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}(\...
- 12.9k
3
votes
0
answers
255
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 ...
- 12.9k