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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.

141 questions
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Is there a less-trivial integer function with described properties?

To be found are integer one-valued functions $f(n_1,n_2)$ with following properties: $f(n_1,n_2)=f(n_2,n_1)$, $f(f(n_1,n_2),n_3)=f(n_1,f(n_2,n_3))$, $f(n_1+n_2,n_1+n_3)=n_1+f(n_2,n_3)$. So far I have found only the functions $\min(n_1,n_2)$ and…
user
  • 27,958
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When does a semiring extend to an integral domain?

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 multiplication on $B$…
Alex
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Is there a name for a ring with a distinguished absorbing element (for both addition and multiplication)?

There's a standard notion of a monoid with zero in algebra, which is a monoid $M$ having a distinguished element $0$ such that $0m=0=m0$ for all $m\in M$. Is there a common name for the ring analogue of this notion, that of a "ring" $R$ with a…
Emily
  • 1,331
5
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Homomorphism of a set to its power set.

Let $(S, +, \cdot, 0)$ and $(S', \oplus, \otimes, 0')$ be two semirings. Then $f: S\rightarrow S'$ is said to be a homomorphism if for all $a, b\in S,$ $f(a+b)=f(a)\oplus f(b)$, $f(a.b)=f(a)\otimes f(b)$ and $f(0)=0'.$ Let $\Bbb Z$ be a set of…
gete
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5
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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 obvious sequence of ring homomorphisms $$\cdots \to…
pregunton
  • 6,358
4
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2 answers

Semirings which cannot be extended to semifields

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 $\cdot$ distributes over $+$. If $(S,\cdot)$ is in fact a…
Antoine de Saint Germain
  • 103
4
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Reference Request for Axiomatic/Algebraic Big $\mathcal{O}$ and Little $o$

I have seen the formal definitions of big $\mathcal{O}$ and little $o$, and do all right working with them. Still, I have some questions that a good reference might help clear up. In what level of generality can $o$ and $\mathcal{O}$ be defined? It…
CTVK
  • 517
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Functions in calculus --- partial functions or functions on context-dependent domain?

Let $S_1,S_2\subseteq \mathbb{R}$. Given two functions $f_1\colon S_1\to \mathbb{R}$ and $f_2\colon S_2\to \mathbb{R}$, we can define a new function $f_1+f_2\colon S_1\cap S_2\to \mathbb{R}$ by the rule $(f_1+f_2)(x)=f_1(x)+f_2(x)$. In this way,…
Pace Nielsen
  • 699
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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: $\mathbf{Q}:=\mathrm{Frac}(\mathbf{Z})$; the $p$-adic integers $\mathbf{Z}_p$ and…
Emily
  • 1,331
4
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1 answer

The spectrum of a semiring

One of the generalizations of algebraic geometry is provided by the theory of semiring schemes, viz. Lorscheid 2012. The theory follows the same set up of scheme theory, but we use semirings instead of rings, also known as rings without additive…
Emily
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4
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1 answer

Are affine semiring schemes equivalent to semirings?

One of the generalizations of algebraic geometry is provided by the theory of semiring schemes, cf. Lorscheid 2012. The theory follows the same set up of scheme theory, but we use semirings instead of rings. Given a semiring $R$, we have a…
Emily
  • 1,331
4
votes
1 answer

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 multiplication over addition to $expr$, that is,…
abebebebahabe
  • 151
4
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1 answer

Does every finitary monad with this propery arise as a free module monad?

Let $T:\mathbf{Set} \rightarrow \mathbf{Set}$ denote a finitary monad such that $T(\emptyset) \cong 1$ and $T(A \sqcup B) \cong T(A) \times T(B)$, naturally in $A$ and $B$. Question. Does $T$ necessarily arise as the free module monad for some…
goblin GONE
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4
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1 answer

What is a semiring without commutativity requirement for addition?

A semiring is defined by two operations $+$ and $\times$ on a set $R$, such that: $(R, +)$ is a commutative monoid $(R, \times)$ is a monoid $\times$ left-distributes and right-distributes over $+$ The identity element for $+$ is an absorbing…
a3nm
  • 664
4
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1 answer

Zeros of a Tropical polynomial

Consider the Tropical semiring $(\mathbb{R}\cup\{-\infty\},\max,+)$. We define $x$ as a zero of a tropical polynomial $f(x)$ if $f$ attains its maximum twice at point $x$ in its linear parts. Why is this consistent with the intuitive idea that we…
AmirHosein Sadeghimanesh
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