# Questions tagged [semiring]

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### Identity elements of semirings

When we define a group the identity element can be any suitable one. E.g. $(\mathbb{N}, 0, +, -)$ or $(\mathbb{N}, 1, \cdot , N_i^{-1})$ are two groups with $2$ different identity elements. Now it is ...
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### Additive and multiplicative identities of a semiring can coincide.

Semiring $(S, \oplus, \otimes, \bar{0}, \bar{1})$ is an algebraic structure in which $(S, \oplus, \otimes)$ is a monoid and $(S, \otimes, \bar{1})$ is a semigroup and multiplication $\otimes$ ...
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### Semirings of small orders

A semiring is a structure $(R, +, 0, *)$ such that $(R, +, 0)$ is a commutative monoid, $(R, *, 0)$ is a semigroup with zero, and the distributive laws hold. I know that there were attempts at ...
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### Additive identity of a semiring and sub semiring.

We know that the identity of a group and its sub groups are same. But I think in case of semirings, the additive identity of a sub semiring may be distinct from that of the semiring. I see a counter ...
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### Can you help me with a semi-ring problem please?

The problem: Let $X = \mathbb{R}$ Define $S = \{ G \subseteq \mathbb{R} : G$ is at most countable $\}$ $\cup$ $\{\emptyset \}$. Show $S$ is a semi-ring. Here is what I have so far. Well the ...
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### Is every idempotent semiring a completely regular?

A semiring $(S, +, \cdot)$ is said to be a completely regular semiring if for every $a\in S$, there exists some $x\in S$ satisfying the following conditions: (1) $a=a+x+a$ (2) $a+x=x+a$ and (...
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Edit: In a semiring $(R, +, \cdot, 0, 1)$, $0$ and $1$ are additive identity and multiplicative identity, respectively such that $0$ is multiplicatively absorbing, that is $0\cdot a=0=a\cdot 0$ for ...
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### Name and significance of this subset of a semiring

Let $(S, \oplus, \otimes)$ be a commutative semiring and $H$ be a subset of $S$ such that for all $x, y\in H$ implies that $x\otimes y= e$ where $e$ is the multiplicative identity( i.e., the ...
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### Uniqueness of the result of rewritting an algebraic expression using distributivity rule

Let $expr$ be an algebraic expression involving natural numbers, addition operator and multiplication operator, e.g., $$(1+2)\cdot(3+4 \cdot 5)+6.$$ By iteratively applying the distributivity of ...
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### Structure of $(a,b)$-adic natural numbers

Introduction One way to define the $n$-adic integers is to start from the ring of modular integers $\mathbb{Z}/n\mathbb{Z}$, the quotient of $\mathbb{Z}$ by the ideal $n\mathbb{Z}$. There is an ...
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### What's more general than a near-semiring?

I'm looking for the names of algebraic structures that are more general than a near-semiring. My preferred structure requires both + and · only be magmas; two semigroups is still over-constraining (...
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### Is the category of semimodules over a semiring abelian?

We consider a semimodule over commutative semiring. It's a well-known fact that the category of $R-$modules (over a ring) is an abelian category. Does this result generalize to the category of $R-$...
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### Multiple definitions of Semirings

I am currently studying for my algebra exam and came across the definition of a semiring. Reading through multiple books at once to better understand the definitions and examples I encountered ...
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### How do you name these semirings?

I came across a family of semirings with no invertible elements. Do these semirings have a special name? Thanks!
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### meaning of the argmax in max-plus semiring (or tropical semiring)

The max-plus semiring:$(\{\mathbb{R}\cup{-\infty}\}, max,+):=(\{\mathbb{R}\cup{-\infty}\}, \oplus, \otimes)$ is defined as an algebraic structure that has max as addition and + as multiplication. My ...
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### What is the significance of the precedence of a semiring/ ring operations?

In general, a semiring/ring is equipped with two distinct binary operations, namely addition $(+)$ and multiplication $(×)$ , where in most of the cases, multiplication distributes over addition and ...
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### Product of two left invertible elements is also left invertible in Semigroup

Consider a Semigroup $(M, \ast)$ with a neutral element $e$. Now I have to prove that all left invertible elements of $(M, \ast)$ form a sub-semigroup. A left invertible element is an element whose ...
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### Left Adjoint to the Forgetful Functor from Rings to Semirings.

Is there a left adjoint to the forgetful functor from the category of commutative rings to the category of commutative semirings? A semiring is like a ring, except for the existence of an additive ...
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### Does an ideal of a ring/semiring necessarily contain identity?

A subset $I$ of a ring/semiring $R$ is said to be an ideal of $R$ if $x+y\in I$ for all $x,y\in I$ and $x.a\in I$($a.x\in I$) for all $a\in R$. Since $0\in R,$ $x.0\in I$ for all $x\in I$ or $0\in I.$ ...
By definition a semiring is zero-sum free if $x\oplus y=\bar{0}$ implies $x=y=\bar{0}$ for all $x,y \in \mathbb{K}$. The tropical semiring is zero-sum free. I'm using this definition for the tropical ...
We know that on an idempotent semiring $R$, the natural order relation is defined as: for all $x, y\in R$, $x\leq y$ when $x+y=y$, which is clearly a partial order relation. I am unable to point out ...