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Questions tagged [semiring]

For questions related to semiring. In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

A semiring is a commutative semigroup under addition and a semigroup under multiplication. A semiring can be empty.

For more on this, check this link and/or this link.

131 questions
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Is every partially ordered semiring an idempotent?

I preferred to call here a semiring $(R, +, .)$ to be an idempotent if $x+x=x$ and $x.x=x~\forall~x\in R$. It is apparent that we can define certain partial order relations on an idempotent semiring. Can we define a partial order relation on a none…
gete
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Let $R$ be an idempotent semiring and $ax=y$ and $by=x~\forall ~x,y\in R$, then show that $x=y$

Here, a semiring $(R, +, .)$ is an idempotent in the sense that $x+x=x$ and $x.x=x~\forall~x\in R$. Let $R$ be an idempotent semiring and $ax=y$ and $by=x~\text{if }x\leq y~\text{and }y\leq x,~\text{respectively }\forall ~x,y\in R$ and some $a, b$…
gete
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What can we say about the subset $\{x \in R \mid x^2 = ax+b\}$ when $R$ is a commutative rig (i.e. a ring without negatives)?

By a rig, I mean a ring whose elements don't necessarily have additive inverses, sometimes called a semiring. I want to ask very broad and potentially very naive question about solving quadratic equations in rig theory. Suppose we're given: a…
goblin GONE
  • 67,744
0
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1 answer

Finitely additive set function on a semiring

$\Omega$ is a set and $\mathcal{C}$ is a collection of subsets of $\Omega$. We call $\mathcal{C}$ is a semiring if and only if $\emptyset\in\mathcal{C}$ and $A,B\in\mathcal{C}\Rightarrow A\cap B\in\mathcal{C}, A\setminus B\in\mathcal{C}_{\Sigma…
Stephen
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Problems about the "sub"$\sigma-$field generated by a set in a semiring.

Suppose $\mathscr S$ is a semiring on $X$, and $\mathscr F=\sigma(\mathscr S)$ is the $\sigma$-field generated by $\mathscr S$. If $A\in\mathscr S$, denote by $\mathscr S_A=\{A\cap B:B\in\mathscr S\}$ and $\mathscr F_A=\{A\cap B:B\in\mathscr F\}$.…
Attendre
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Name this algebraic structure

I found an algebraic structure $(P(S), \cup, \cap)$, where $S$ is a set of some elements and $P(S)$ is its power set such that the given algebraic structure satisfies all the properties of a semiring except distributivity. When it comes to…
gete
  • 1,352
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What is the necessary and sufficient condition for a subset of the semiring to be sub- semiring

Is there any smaller sort of things (without proving semiring axioms ) to ascertain that a subset $R$ of the semiring $S$ is sub-semiring ? Or, What is the necessary and sufficient condition for a subset of the semiring to be sub- semiring ?
gete
  • 1,352
0
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1 answer

Can both additive and multiplicative operations in a semiring distribute over each other?

In general, the multiplicative operation in a semiring distributes over the additive operation from both the left and the right. But i found some operations which satisfy all the conditions of a semiring structure in which the multiplicative…
gete
  • 1,352
0
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2 answers

Why do we need left distibutivity when using semirings for shortest-path problems?

This paper describes a framework for shortest-path algorithms based on semirings. My question is, why do we need left distributivity of ⊕ (generalized sum) over ⊗ (generalized product) for the algorithms to give correct results. In other words, why…
skvadrik
  • 101
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Why semirings can not have the additive inverse and canonical partial order?

In a computer science class about automata the professor claims that a semiring can not have both additive inverse and canonical partial order at the same time. I do not understand how additive inverse contradicts with canonical partial order? The…
viefs
  • 101
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1 answer

how do we prove the following equality in idempotent semiring

Let $(G,.,e)$ be a torsion-free group with a total order that is compatible with the group operation, by which we mean that if $a,b \in G$ with $a \leq b$ then $ca \leq cb$ and $ac \leq bc$ for all $c\in G$. We can turn $G$ into an idempotent…
Petchimuthu Subramanian
  • 1
0
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1 answer

near-semirings related problems

If $(S, +,.)$ is a distributive near-semiring, then prove that the subsemigroup $(S^2, +)$ is additive subcommutative. [ Hints: A non empty set $S$ with two binary operations $'+'$ and $'.'$ is called near-semirirng if the following conditions are…
Petchimuthu Subramanian
  • 11
-2
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1 answer

Semirings related

How to prove a distributive near-semiring $(S,+,.)$ with multiplicative identity $1$ is additive subcommutative? [ Hints: A non empty set $S$ with two binary operations $'+'$ and $'.'$ is called near-semirirng if the following conditions are…
Petchimuthu Subramanian
  • 11