Questions tagged [semirings]
46 questions
3
votes
3 answers
($\oplus$, $\otimes$) 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 max and softmax (logsumexp) are solutions. Can we characterize all possible…
rafaelcosman
- 133
3
votes
0 answers
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 examples where it is:
Obviously, if…
Mike Shulman
- 65,064
2
votes
1 answer
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 just mean a function $*:S \rightarrow S$…
2
votes
1 answer
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
$(+,\otimes_{-\infty})$ is max-plus tropical semiring on…
Tegiri Nenashi
- 244
1
vote
0 answers
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 a ring, it should usually be a lattice. More…
