Questions tagged [semiring]

Question 1

For the case of commutative field, we have that each vector space has a basis and even more, each linearly independent set can be completed into a basis. Do we have a same result for general ...

Question 2

I usually work with lattices, and one of my professors specializes in this topic as well. However, in the course I am taking, we focus on classical concepts from module theory, rings, and fields, ...

Question 3

Let $A$ and $B$ be $n \times n$ matrices with integer coefficients. Consider the tropical multiplication of matrices given component-wise by $$(A\otimes B)_{ij} = \max_{d \in [n]} \left\{ a_{id} + b_{...

Question 4

For the purpose of this post, a semiring is an algebraic structure satisfying the axioms of a (unital) ring except the existence of additive inverses. Recall the equivalent characterizations of a ...

Question 5

I am trying to prove the following lemma (originally Prop. A.2) which I found from the paper "Containment of conjunctive queries on annotated relations" by Todd J. Green. Note that although ...

Question 6

Prove that a commutative ring $R$ is a semiprimitive ring if and only if $0$ is the only nilpotent element in $R$. Attempt: I start by noting that a ring is semiprimitive if its Jacobson radical $J(R)$...

Question 7

Definitions By a commutative $\textit{semiring}$ (with 1 and without 0), I mean a triple $(S,+,\cdot)$ where $(S,\cdot)$ is a commutative monoid, $(S,+)$ is a commutative semigroup, and $\cdot$ ...

Question 8

I'm looking into semirings. It is known that is possible to encode derivation over a semiring by considering $d(a+b) = d(a) + d(b)$ and $d(a\cdot b) = d(a) \cdot b + d(b) \cdot a$. Both operations are ...

Question 9

Consider the set of length $n$ Boolean vectors, with addition defined as component-wise OR, and multiplication by a boolean scalar value defined by component-wise AND with that scalar, as expected. ...

Question 10

A commutative semiring $R$ with exponentiation has an additional operation $R \times R \to R, (a, b) \mapsto a^b$ (called exponentiation), satisfying the following six axioms ($\forall a, b, c \in R$):...

Question 11

A famous result of Funayama and Nakayama states that the congruence lattice of any lattice is a distributive lattice [1]. Also, it can be proved that the lattice is a frame/ complete Heyting algebra 2....

Question 12

Some background The natural numbers $\mathsf{Nat}=\{0,1,2,\dots\}$ has the structure of a semiring under addition and multiplication. Write $\mathsf{HFS}$ for the set of all hereditarily finite sets (...

Question 13

I know the group completion of a monoid. If we have a semiring $R$, which is in particular a monoid. Then the group completion of $R$ is an abelian group. But how can we define the completion of a ...

Question 14

The injective continuous map $h:X\to Y$ is a topological embedding if it is a homeomorphism onto the image of $h$ in $Y$. Let the function $h:X\to Y$ be a topological embedding and let $X=Y$ be a ...

Question 15

Mirroring the construction of $\mathbb{Z}$ from $\mathbb{N}$, we can extend a commutative and additively cancellative semiring $A$ to its additive group of differences, $B$, and then define ...

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Questions tagged [semiring]

Semivector spaces over commutative semifields

Connection Between Bounded Distributive Lattices and Module Theory: Applications Linking Both Areas?

Hardness of tropical matrix factorization

Semiring with a unique left ideal

Surjective homomorphism of semirings is an order isomorphism?

Prove that a commutative ring $R$ is a semiprimitive ring if and only if $0$ is the only nilpotent element in $R$

Semirings which cannot be extended to semifields

Integration over a semiring

Maximal number of independent vectors in a Boolean space

Reference for semirings with exponentiation

Congruence lattice of a semiring

Relation between semiring structure of natural numbers and their hereditarily finite sets structure under bitwise operations

definition of group completion of semirings

Is there a name for a topological embedding (of a semiring) into itself?

When does a semiring extend to an integral domain?

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