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.

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Relation between semiring structure of natural numbers and their hereditarily finite sets structure under bitwise operations

Some background The natural numbers $\mathsf{Nat}=\{0,1,2,\dots\}$ has the structure of a semiring under addition and multiplication. Write $\mathsf{HFS}$ for the set of all hereditarily finite sets (...
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definition of group completion of semirings

I know the group completion of a monoid. If we have a semiring $R$, which is in particular a monoid. Then the group completion of $R$ is an abelian group. But how can we define the completion of a ...
Ziqiang Cui's user avatar
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Is there a name for a topological embedding (of a semiring) into itself?

The injective continuous map $h:X\to Y$ is a topological embedding if it is a homeomorphism onto the image of $h$ in $Y$. Let the function $h:X\to Y$ be a topological embedding and let $X=Y$ be a ...
it's a hire car baby's user avatar
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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 ...
Alex's user avatar
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How many affine prime-quotient ultrafilters does a rational semiring have?

I know ultrafilters are considered powerful by more-learned mathematicians than I. I cannot profess to understand the reasons how and why although I can see the power of Zorn's Lemma and the axiom of ...
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Help with identification of a semiring with an "almost-inverse".

I'm working with an idempotent semiring which have families $C^v, \hat{C^v}$ of elements with the following properties: $$ {C}^v_i \hat{C_i^v} = 1 $$ $$ \sum_i \hat{C_i^v} {C}^v_i = 1 $$ $$ {C}^v_i \...
Łukasz Lew's user avatar
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1 answer
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Given the sums by line and by column, find the corresponding matrix (in a general (complete ?) monoid or semiring)

I'm interested in the following question: In a monoid $\mathbf{M}$, given sums $\sum_{i \in I} a_i = \sum_{j \in J} b_j$, is it possible to find elements $c_{i,j}$ such that for all $i$, $a_i = \...
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Classification of graded fields, semifields, and skew fields

Recently I came upon the following result: Result 1. Let $K$ be a $\mathbb{Z}$-graded field. Then either $K$ is trivially graded (i.e. $K_k=0$ for $k\in\mathbb{Z}\setminus\{0\}$ with $K_0$ a field or ...
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The free semiring on the Boolean monoid

Recall the following definitions: The Boolean monoid is the monoid $\mathbb{B}=(\{0,1\},\max,0)$; The free semiring on a commutative monoid $A$ is the semiring $\mathrm{Free}(A)$ consisting of The ...
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Is there a name for the exponential semiring of G-sets?

Is there a name for the exponential semiring of G-sets? In an ordinary type system, the types are like sets. It's possible to extend this analogy and get a $G$-type system where every type is acted on ...
Greg Nisbet's user avatar
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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 ...
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Classification of "complete dense ordered near-semirings"

Let's define a complete dense ordered near-semiring, or a CDON, as a set $A$ equipped with two binary operations $+,\times$ and a binary relation $\leq$ such that: $+$ is a monoid, whose identity is ...
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Sigma-algebra generated by collection of cylinders.

Let $X$ = {$0,1$}$^\mathbb{N}$ be the set of all infinite sequences of $0$’s and $1$’s. A typical element $x \in X$ is written as $x = x_1x_2x_3$···. A cylinder set is a subset of $X$ of the form {$x \...
Bob's user avatar
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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 $\...
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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 ...
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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=...
gete's user avatar
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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 ...
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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\...
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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 ...
Pace Nielsen's user avatar
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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}(\...
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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 ...
Math101's user avatar
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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 ...
Emily's user avatar
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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$^\dagger$...
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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 ...
Emily's user avatar
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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 ...
Emily's user avatar
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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(\...
Akshara Prasad's user avatar
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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 ...
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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 ...
gete's user avatar
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1 answer
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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 ...
gete's user avatar
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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 ...
Brendan Mallery's user avatar
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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 ...
deaton.dg's user avatar
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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 ...
gete's user avatar
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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 ...
Jim's user avatar
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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 $\otimes$ ...
gete's user avatar
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2 votes
2 answers
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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 ...
gete's user avatar
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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, ...
gete's user avatar
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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://...
Alex C's user avatar
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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 ...
Lehs's user avatar
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1 answer
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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 ...
gete's user avatar
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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?
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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 ...
gete's user avatar
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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 $\mathscr S_A=\{A\cap B:B\in\mathscr S\}$ ...
Attendre's user avatar
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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 ...
gete's user avatar
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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 ...
Boogie's user avatar
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$\alpha \subset P(\Omega) : A, B\in \alpha \Rightarrow A\cup B \in \alpha, A \cap B \in \alpha$ 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 ...
CodeSun's user avatar
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1 answer
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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 ...
Overachiever's user avatar
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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=(...
gete's user avatar
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1 vote
1 answer
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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 $\...
gete's user avatar
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1 vote
1 answer
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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 $\...
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