Questions tagged [semiring]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
1 answer
67 views

Pre-image of a semi-ring is not a semi-ring. Measure Theory

I'm studying measure theory using the book "Introduction to Measure and Integration", from S.J. Taylor. There, in section 1.5 Classes of subsets, he defines a Semi-ring as: "A class $\...
user avatar
  • 89
0 votes
0 answers
33 views

Applications of using elements of $k$-closed commutative semirings as edge-weights in directed graphs

With much interest I have been reading Mehryar Mohri's paper on "Semiring Frameworks and Algorithms for Shortest-Distance Problems". In this paper Mohri describes a generalisation of the ...
user avatar
  • 117
1 vote
1 answer
24 views

Connection between kernels of linear maps of semimodules and injectivity

Let $S$ be a semiring (i.e. satisfies all the ring axioms besides existence of additive inverses) and $M, N$ semimodules over $S$ (same thing). For a linear map $\varphi : M \rightarrow N$, we can ...
user avatar
  • 402
0 votes
0 answers
31 views

What are conditions on a ring $R$ for all finitely generated submodules of $R^n$ with torsion-free quotients to be free?

Let $R$ be a commutative integral ring, and let $M$ be a finitely generated sub-module of $R^n$ such that $R^n / M$ is torsion-free (that is, if $rx \in M$ then $x \in M$ for all $x \in R^n$ and $r \...
user avatar
0 votes
0 answers
14 views

Difference Ternary and gamma

Difference between ternary semiring and gamma semiring. There are many articles found on both topics. But what is the major difference between ternary and gamma
user avatar
-1 votes
1 answer
24 views

Set of elements of an idempotent semirings are totally ordered.

An element $S$ is said to be totally ordered set if for all $a, b\in S\implies$ either $a\leq b$ or $b\leq a.$ An algebraic structure $(S, +, \cdot)$ is said to be an idempotent semiring if $x\cdot x=...
user avatar
  • 1,174
0 votes
0 answers
46 views

Prove relations for a content $\mu: \mathcal H \to \mathbb R$ where $\mathcal H$ is a seminring over the set $\mathit X$

I have the following three relations to show in my measure theory exercise course. Let $\mu: \mathcal H \to \mathbb R$ be a content on the semiring $\mathcal H$ over the set $\mathit X$ and $\mu^{*}$ ...
user avatar
  • 166
3 votes
2 answers
68 views

Rig in which 0 is not an absorber.

I've been trying to find a rig-like structure (a set $R$ with a monoid structure $(R,\cdot,1)$ and a (commutative) monoid structure $(R,+,0)$ such that the multiplication distributes over addition) in ...
user avatar
  • 179
0 votes
1 answer
27 views

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\...
user avatar
  • 387
0 votes
0 answers
30 views

What is minimal semiring, ring and algebra?

I know what semiring, ring, and algebra mean. $S$ - semiring: $∅ ∈ S$ $∀ A ∈ S, ∀ B ∈ S -> A ∩ B ∈ S$ $∀ A ∈ S, ∀ A_1 ∈ S, A_1 ⊂ A -> ∃ A_2, …, A_n ∈ S : A_1 ⊔ A_2 ⊔ … ⊔ A_n = A$ $R$ - ring: $...
user avatar
4 votes
0 answers
35 views

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 ...
user avatar
4 votes
0 answers
134 views

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}(\...
user avatar
  • 421
0 votes
1 answer
50 views

Well-ordered commutative semirings

I am interested in the characterization of the most important types of numbers from an axiomatic viewpoint. For example, every complete ordered field is isomorphic to the field of real numbers. In ...
user avatar
  • 800
4 votes
1 answer
81 views

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 ...
user avatar
  • 421
2 votes
0 answers
59 views

Is there a closed form for the recursively defined function $f_{n+1}(x) := f_n(x)^{f_n(x)}$.

Is there a closed form$^*$ for the following recursive function: \begin{align} f_0(x) &:= x\\ f_{n+1}(x) &:= f_n(x)^{f_n(x)} \end{align} Assuming that $x$ is an element of a semi-ring$^\dagger$...
user avatar
  • 23
3 votes
1 answer
112 views

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 ...
user avatar
  • 421
3 votes
1 answer
138 views

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 ...
user avatar
  • 421
0 votes
0 answers
27 views

Prove that there exists a sequence of sets of finite measure in a $\sigma$-algebra that increases to the entire set.

Let $\mathcal{A}$ be a semiring on a set $\Omega$. And let $\mu$ be a measure on $\sigma(\mathcal{A})$ that is $\sigma$-finite on $\mathcal{A}$. Prove that there exists a sequence of sets in $\sigma(\...
user avatar
1 vote
1 answer
26 views

Is there a name for integer division in semirings?

For a rig (or semiring) $R$ we can define for $n \in \mathbb{N}$ an element $n * x = \underbrace{x + \dots + x}_n$. Is there a standard name for the property that $n * x = n * y$ implies $x = y$ for ...
user avatar
  • 339
1 vote
1 answer
29 views

A property of semiring

Definition 1. Let $(R, +, \cdot)$ be a semiring equipped with a relation $\leq$ such that the relation is defined as: for all $x, y \in R$, $x\leq y$ if and only if $x+a=y$ (or $a+x=y$) for some $a\in ...
user avatar
  • 1,174
0 votes
1 answer
27 views

Property of a semiring equipped with partial order relation [closed]

Let $R$ be a multiplicatively idempotent semiring with additive identity, and a partial order relation $\leq$ is defined on $R$. Then, for all $x$ in $R$, does the identity $x+2x=2x$ implies $x\leq ...
user avatar
  • 1,174
2 votes
0 answers
46 views

How many tropical polynomials give rise to the same variety? (reference request)

An n-variable polynomial $p(x)=\bigoplus_{i=1}^n \beta_i \textbf{x}^{\alpha_i}$ (here $\textbf{x}$ is the tuple $(x_1,x_2,...,x_n)\in \mathbb{R}^n$) in the tropical (max-plus) semiring has an ...
user avatar
1 vote
0 answers
45 views

If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital?

Suppose $S$ is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring $G(S)$ is a unital ring. Can we conclude that $S$ must be ...
user avatar
  • 138
1 vote
0 answers
24 views

Difference between subtractive and k- ideal of a semiring.

An ideal $I$ of a semiring $S$ is said to be k-ideal if $a\in I$ and $x\in S$, and if either $a+x\in I$ or $x+a\in I$ then $x\in I$. An ideal $I$ of a semiring $S$ is said to be $subtractive$ if $a\in ...
user avatar
  • 1,174
0 votes
2 answers
53 views

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 ...
user avatar
  • 1,551
1 vote
0 answers
28 views

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 $\otimes$ ...
user avatar
  • 1,174
2 votes
2 answers
61 views

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 ...
user avatar
  • 5,773
1 vote
0 answers
38 views

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 ...
user avatar
  • 1,174
0 votes
1 answer
41 views

Simplify $a.b+c.d$

Suppose that, $(S, +, \cdot)$ is a semiring, where the operations are defined as $x\cdot y=min(x, y)$ and $x + y=max(x, y)$. Can we further simply the expression $a\cdot b+c\cdot d$?, where $a, b, c, ...
user avatar
  • 1,174
1 vote
1 answer
175 views

Additive zero of a semiring

The Encyclopedia of Mathematics states that a semiring is "A non-empty set S with two associative binary operations + and $\cdot$, satisfying the distributive laws"... https://...
user avatar
  • 953
-1 votes
2 answers
121 views

A logic better adapted to quantum phenomena?

Our way of mathematical thinking is totally controlled by a simple two-valued logic $(\mathbf{false}, \mathbf{true})$. All deductions are due to this logic and we are unable to think otherwise. But ...
user avatar
  • 13.3k
-4 votes
1 answer
59 views

prove that $x=ax+b$ has solution for all values of $a$ and $b$ [closed]

Let $\Bbb R_+$ be the set of positive real numbers (including $\infty$). $(\Bbb R_+, +, \cdot, \leq)$ is an ordered semiring endowed with usual addition and multiplication in $\Bbb R_+$. Then it is ...
user avatar
  • 1,174
1 vote
0 answers
90 views

direct product decompositions of semi-rings

(With Hilbert's Basis Theorem in mind.) Is it true that every finitely presented commutative semi-ring with unit is a finite direct product of directly indecomposable factors?
user avatar
  • 185
0 votes
0 answers
24 views

Application of semiring in data (or signal) flow

Suppose certain input data (or signals) have to flow from the source (or sender) $A$ to the target (or receiver) $B$ through a node (or station) $S$. Suppose that at an instant $t$, $S$ being a set of ...
user avatar
  • 1,174
0 votes
1 answer
19 views

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 $\mathscr S_A=\{A\cap B:B\in\mathscr S\}$ ...
user avatar
1 vote
0 answers
28 views

Does every idempotent semiring has characteristic zero?

Let $(S, +, \cdot, 0, 1)$ be a semiring. If for all $s\in S$ and non- negative integer $m$, $ms=s+s+...+s(m$times) $=0$, then $S$ is said to be of characteristic $m$. If no such $m$ exists then $S$ is ...
user avatar
  • 1,174
0 votes
0 answers
74 views

Invertible elements in the ring completion of a semiring.

Let $R$ be a commutative semiring with unit. Assume that addition is cancellative so that $R$ is included in its ring completion. Fix ${a, b \in R}$. A simple calculation shows that ${a - b}$ is ...
user avatar
  • 185
2 votes
1 answer
31 views

$\alpha \subset P(\Omega) : A, B\in \alpha \Rightarrow A\cup B \in \alpha, A \cap B \in \alpha$ algebra or semiring?

Is it easy to see if $\alpha $ is an algebra or a semiring? I have $\alpha \subset P(\Omega) $ ( $P(\Omega)$ powerset of $\Omega$ ) where $ A, B\in \alpha \Rightarrow A\cup B \in \alpha, A \cap B \in ...
user avatar
  • 31
1 vote
1 answer
77 views

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 ...
user avatar
1 vote
0 answers
18 views

Can we define a mapping from the set of graphs to the set of adjacency matrices?

Graph union: the union of two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ is defined and denoted by $G_1\cup G_2=(V_1\cup V_2, E_1\cup E_2)$. Graph intersection: the intersection of two graphs $G_1=(...
user avatar
  • 1,174
1 vote
1 answer
61 views

A semiring isomorphic to a cartesian product of two semirings

Let $S=\lbrace 0, 1, 2, 3, 4, 5 \rbrace$, $R=\lbrace 0, 1\rbrace$ and $T=\lbrace 0, 1, 2\rbrace$, and the addition (+) and the multiplication $(\cdot)$ in $S$, $R$ and $T$ be defined by $\max$ and $\...
user avatar
  • 1,174
1 vote
1 answer
36 views

Image of a semiring homomorphism, which is an ideal.

Let $\phi: (R, +, \cdot)\rightarrow (S, \oplus, \star)$ be a semiring map. Then $\operatorname{im}\phi$ is a sub semiring of $S$. In general, it is not an ideal. I look for an example in which $\...
user avatar
  • 1,174
0 votes
0 answers
47 views

What does the kernel of a semiring homomorphism being a null set signify?

According to wikipedia, the kernel of a homomorphism measures the degree to which a homomorphism fails to be injective. What does it mean when a kernel of semiring is a null set? That is when $f:S\...
user avatar
  • 1,174
0 votes
1 answer
45 views

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 (...
user avatar
  • 1,174
2 votes
1 answer
137 views

Are additive zero and additive identity the same thing?

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 ...
user avatar
  • 1,174
1 vote
1 answer
80 views

Ideals of a ring/semiring

Define the ideals of a semiring as follows: A non empty subset $I$ of a semiring $S$ is said to be a left (resp. right) ideal of $S$ if: (1) $a+b\in I$ for all $a, b\in I$ and (2) $sa$ (resp. $...
user avatar
  • 1,174
5 votes
2 answers
258 views

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(...
user avatar
  • 1,174
0 votes
1 answer
12 views

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 ...
user avatar
  • 1,174
0 votes
0 answers
17 views

About distributivity of a semiring

I know that if $(R, +, ., 0, 1)$ is an idempptent semiring and $x, y\in R,$ then by distributivity of $\cdot$ over $+$, we can write $x+xy+x=x(1+y+1)=x.1=x.$ This time, if my semiring is $(P(X), \cup, ...
user avatar
  • 1,174
0 votes
0 answers
27 views

Number of elements in a set that forms a semiring

I wonder wether there is any possible rule to determine the number of elements in a set $ S$ under certain binary operations if it forms a semiring or, can we find the number of elements in a set ...
user avatar
  • 1,174