Questions tagged [semiring]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
0answers
22 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(\...
1
vote
1answer
21 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 ...
1
vote
1answer
23 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 ...
0
votes
1answer
22 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 ...
0
votes
0answers
10 views

Property of an ordered semiring

If $(R, +, \cdot \leq)$ is an ordered semiring, then for all $x,y\in R$, $x\leq y$ implies that there exists $a\in R$ such that $a+x=y$ and $bx=y$ for some $a, b\in R.$ If $1+a=b$ for $a, b\in R$. ...
2
votes
0answers
32 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 ...
1
vote
0answers
36 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 ...
1
vote
0answers
10 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 ...
0
votes
2answers
36 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 ...
1
vote
0answers
23 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 $\otimes$ ...
1
vote
2answers
48 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 ...
1
vote
0answers
28 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 ...
0
votes
1answer
40 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, ...
1
vote
1answer
145 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://...
-1
votes
2answers
117 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 ...
-4
votes
1answer
56 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 ...
1
vote
0answers
85 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?
0
votes
0answers
19 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 ...
0
votes
1answer
16 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 $\mathscr S_A=\{A\cap B:B\in\mathscr S\}$ ...
1
vote
0answers
20 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 ...
0
votes
0answers
62 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 ...
2
votes
1answer
30 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 ...
0
votes
1answer
52 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 ...
1
vote
0answers
10 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=(...
1
vote
1answer
44 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 $\...
1
vote
1answer
29 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 $\...
0
votes
0answers
26 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\...
0
votes
1answer
36 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 (...
2
votes
1answer
77 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 ...
0
votes
1answer
38 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. $...
5
votes
2answers
198 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(...
0
votes
1answer
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 ...
0
votes
0answers
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, ...
0
votes
0answers
26 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 ...
0
votes
1answer
523 views

Is the set of edges a graph a subset of that graph?

Let $G=(V, E)$ be a non empty and undirected simple graph with its edge set $E$. Can we say that $E$ is a subset or sub graph of $G$ in any sense i.e., $E\subseteq G$ ? If so, then we can enjoy some ...
0
votes
1answer
15 views

Regular semiring (in the sense of von Neumann)

A semiring $R$ is called regular (in the sense of von Neumann) if for each $a\in R$, there exists $x, y\in R$ such that $a+axa=aya.$ Reference here Further, from the above link, I encounter that ...
1
vote
1answer
35 views

About the definition of a (completely) regular semiring

I see two definitions of a regular semiring. $1^{st}$ definition: A semiring $R$ is called a regular semiring if for every $x\in R$; $x = xax$ for some $a\in R$. $2$nd definition: A semiring $(R, +,...
3
votes
1answer
65 views

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 ...
3
votes
0answers
80 views

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 ...
1
vote
1answer
79 views

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 (...
1
vote
1answer
110 views

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 $R-$...
1
vote
1answer
62 views

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 ...
0
votes
1answer
46 views

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!
0
votes
0answers
30 views

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 ...
2
votes
2answers
81 views

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 ...
0
votes
1answer
51 views

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 ...
2
votes
1answer
65 views

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 ...
0
votes
2answers
22 views

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 $0\in I.$ ...
0
votes
1answer
84 views

Why can't the tropical semiring be zero-sum free without both x and y equaling 0?

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 ...
1
vote
2answers
69 views

Is the natural order relation on an idempotent semiring total/linear?

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 ...