# Questions tagged [monoid]

A monoid is an algebraic structure with a single associative binary operation and an identity element.

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### Equivalence relation in construction of Grothendieck group

Sorry if this has been asked before but I couldn't find the question I have. Yesterday I read the wikipedia page for a Grothendieck group. It provided two explicit constructions given a commutative ...
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### A weak cancellation property for monoids

Suppose $M$ is a (commutative) monoid. Typically the cancellation property is defined as $a + c = b + c \Rightarrow a = b$ for all $a,b,c \in M$. Recently I was working on a problem where I thought I ...
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### Converse to a proposition on divisors in commutative monoids

Let $(M,*,1)$ be a commutative monoid. Define the binary relation $R$ on $M$ by $aRb$ iff there exists an $x$ in $M$ such that $a*x=b$. $R$ is the "divides" relation. Since $M$ is a ...
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### Comparability with identity of an ordered semigroup

It is possible to compare any ordered semigroup with $0$: Comparability with zero of an ordered semigroup Let's say an ordered semigroup $S$ is comparable with identity if it can be embedded into an ...
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### If the monoid algebra $R[M]$ is finitely generated, then $M$ is a finitely generated monoid.

Consider a commutative, cancellative, torsion-free monoid $M$ and a commutative ring $R.$ If the monoid algebra $R[M]$ is finitely generated as an $R$-algebra, then $M$ is finitely generated. I am ...
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### Ordered monoid with partial subtraction: Is there a name for this?

I've come across the following structure as being useful to represent various kinds of resources (e.g., fungible things like money, nonfungible things like deeds, etc.), and I was curious if there's a ...
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### What is the identity in the power set of $\Sigma^*$ as a monoid?

Given an alphabet $\Sigma$, $P (\Sigma^*)$, the power set of $\Sigma^*$, is a monoid, with language concatenation as morphism. What is the identity: the empty language, or the language consisting of ...
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### Does string substitution have a definition, similar to the one for string homomorphism in terms of monoid morphism of the free monoid?

string homomorphism is defined in formal language theory as: A string homomorphism (often referred to simply as a homomorphism in formal language theory) is a string substitution such that each ...
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### Description of Multicones with Farey intervals

I have a problem in computing and understanding in description of multicones with Farey intervals. Let $M^{*}$ be the monoid On the generators $F_{+}$ , $F_{-}$ operating on words in $A,B$ by the ...
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### Given any equivalence $\sim$ on $X$ do functions $f:X\to X$ such that $f(u)=f(v)\iff u\sim v$ have a specific name?

Given any equivalence $\sim$ on $X$ do functions $f:X\to X$ such that $f(u)=f(v)\iff u\sim v$ have a specific name? I know if I replace the image of each element with its pre-image then this is ...
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### What is the correct name for a “product function” on a monoid?

Let $W$ be a monoid. A function $f\colon W\rightarrow W$ is a "product function" if $f(w)$ is a product of constants in $W$ and positive integer powers of $w$. It could also be called a "non-...
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### Is co/complete category of monoid (or commutative monoid)?

Let $Mon$ (resp., $CMon$) be a category of monoids (resp., commutative monoid) whose morphisms are usual monoid homomorphisms (resp., commutative monoid homomorphism). Is this category complete and/or ...
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### Factorization of the identity element of the free monoid

Let $X$ be a set and let $\text{Mo}(X)=\bigcup_{n\in\mathbb{N}} X^{[1,n]}$. Then $\text{Mo}(X)$ together with the law $(w,w')\mapsto ww'$, where $ww'$ denotes the juxtaposition of the sequences $w$ ...
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### The empty sequence is the identity element of the free monoid constructed on a set X

Let $X$ be a set. Let $w=(x_i)_{1\leq i\leq m}$ and $w'=(x'_j)_{1\leq j\leq n}$ for some $m,n\in\mathbb{N}$. The composition of $w$ and $w'$, denoted by $ww'$, is the family $(y_k)_{1\leq k\leq m+n}$ ...
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### Quotients of semilattices (?)

I want to build some structure and I think a semilattice is the best way to do so, but don't know for sure. I have some (non-empty) set $X$ with some distinguished elements $a,b,c,d$ and $e$. I know ...
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### Each magma $M$ is associated with monoids $\mathcal{L}(M)$ and $\mathcal{R}(M)$. What are these called, and have they been studied?

Let $X$ denote a magma. Then $\mathrm{List}(X)$ is a monoid equipped with both a left and a right action on $X$, where the actions are defined in the obvious way. To illustrate these actions, suppose ...
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### Group congruences: If the operation is preserved, do we get $a\sim b$ $\Rightarrow$ $a^{-1}\sim b^{-1}$?

Let $(G,*)$ be a group. Let $\sim$ be an equivalence relation such that $$(\forall a,a',b,b'\in G)a\sim a', b\sim b' \Rightarrow a*b\sim a'*b'. \tag{*}$$ I.e., the equivalence relation $\sim$ respects ...
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### Commutative and additive monoids [closed]

I am unfamiliar with higher levels of topology and number theory but find myself working on a project in which I need an understanding of some of those topics. Is anyone able to provide information on ...
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### refering something as “non-linear” when there is no underlying linear structure

Can I talk about a non-linear shape functional. I understand a shape functional $J$ as some mapping that takes a shape and returns a real (or complex) value. I would like to talk about a non-linear ...
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### generators and monoid homomorphisms

Is it true that any homomorphism $f: M \to N$ between two monoids $M$ and $N$ maps generators of $M$ to generators of $N$? I am having trouble proving it to myself.
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### Monoid morphisms

For one of my computer science classes, I am asked to give examples of monoid morphisms for the following morphisms. However, I really don't know how to approach it since we haven't worked with ...
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### Monoid that is idempotent induces partial ordering

Given a commutative monoid $(M,0,\oplus)$. Then we can define an ordering on $M$ by $a\geq b :\Leftrightarrow \exists c: a=b\oplus c$. The relation is then transitive and reflexive. The claim is now ...
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### Exponents in $\mathbf{Sets}^{G^{{\rm op}}}$ for an arbitrary group $G$.

This is Exercise I.5(b) of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]." According to Approach0, it is new to MSE. The Details: From p. 17 ibid. . . . Definition 1: ...
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### proving result from monoid theory

An element $x$ of a monoid $M$ is called an atom if $x \neq 1$ and $x=ab$, for some $a,b \in M$, implies that $x=a$ or $x=b$. Furthermore, $M$ is said to be atomic if every element $a \in M$ can be ...
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### The category of monoid objects

Let $C$ be a cartesian category. We can form the category $\text{Mon}(C)$ of monoid objects in $C$. If $C$ is $\text{Set}$, this category is the category of monoids. If $C$ is $\text{Mon}$ (the ...
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### How does Universal Mapping Property encode “no-junk” and “no-noise” in free monoid?

I am going trough the "Category Theory" book by Steve Awodey. In the "1.7 Free categories" chapter the author introduces the following algebraic definition of free monoid: A ...
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### Construction of enveloping group of a monoid

Let $M$ be a monoid and let $G$ be the group which it generates. $G$ can be described as the group obtained from $M$ by adjoining formal inverses. Despite this simple description, I am trying to ...
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### Group freely generated by monoid

There are several ways to define the group freely generated by a monoid, all of which (necessarily) produce isomorphic groups. One way starts with a presentation of the monoid, and simply reinterprets ...
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### Proving Transitivity for the equivalence relation: $\langle a,b \rangle \sim \langle c,d \rangle \iff a+_{\mathbb{N}}d=b+_{\mathbb{N}}c$

For any $a,b,c,d \in \mathbb N$, I am trying to demonstrate that $\langle a,b \rangle \sim \langle c,d \rangle \iff a+_{\mathbb{N}}d=b+_{\mathbb{N}}c$. I am trying to do this without invoking the ...
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### Unique function whose product is a unique monoid element?

Let $P$ be the free monoid on a set $X$. Given $p \in P$, this induces a unique function $f : [n] \to X$ with $\prod_{i\in[n]} f(i) = p$. Does this function have a name?
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### Formula for $M_{p^k} = \{ x \in \Bbb{Z} : x^2 = 1 \pmod {p^k}\}$?

Let $M_n = \{ x \in \Bbb{Z}: x^2 = 1 \pmod n\}$. It is a multiplicative submonoid of $\Bbb{Z}$. When $\gcd(a,b) = 1$ then we have: $$x^2 = 1\pmod {a,b} \iff \\ x^2 = 1\pmod {ab}$$ by the Chinese ...
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### Quotienting by kernel of a monoid homomorphism in the integers.

If $f : \Bbb{Z}^{\times} \to (\Bbb{Z}/n)^{\times}$ is a monoid hom, then is $\Bbb{Z}^{\times} / \ker f \approx \operatorname{im} f$? For example $f(x) = x^2$. That is can we quotient a monoid by the ...
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### Group of mappings

Is there a group $G$ of mappings $X \to X$ that has a non-bijective map in it? I mean, for each element of G, it must has its inverses at right and left, and those must be the same, so the element is ...
Let $E$ be a monoid. I know that the intersection of any family of submonoids of $E$ is again a submonoid of $E$. Under what conditions is the union of an arbitrary family of submonoids of $E$ a ...
### There is no natural polynomial map (other than $1$) that can eventually leave the $k$-semiprimes behind.
Let $F(S)$ be the free commutative monoid on countably many symbols $S$. Then it's obvious that $F(S) = \{1\} \uplus S \uplus S^2 \uplus \dots$ One can take $S =$ the prime numbers in $\Bbb{N}$ in ...