Questions tagged [monoid]

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

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Commutative monoids with "bottom"

Is anyone aware of any existing terminology (and/or research) on the topic of the following structure, like where else it might appear, or what constraints can be placed on f for commutativity : ...
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Is every associative $n$-ary operation with an identity element induced by a monoid?

Given any $n$-ary operation $*$ on a set $X$, an identity element for $*$ is an element $e \in X$ such that $x*e*e*...*e=e*x*e*e*...*e=...=e*e*...*e*x=x$ ($n-1$ $e$s in each product) for all $x \in X$....
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Can you, given any semigroup, define an identity element to make it a monoid

I'm wondering if I can "make up" an identity element, like so: I can define an element I such that any element x + I is equal to x, i.e.: I can redefine my set as [the old set] union with {I}...
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Notation for the product of all elements of a finite commutative subset in a monoid. [duplicate]

Let $M = (A,*)$ be a monoid, $fcs(M)$ be the set of all commutative subsets of $M$ (a subset $S$ of $M$ is commutative iff for all $a,b \in S$ holds $a*b=b*a$). Let $S \in fcs(M)$. Is there a ...
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the amalgamated coproduct $A(I)$ is regular torsionless if $I = (eS]$.

Let $I$ be a right ideal of a pomonoid $S$, $x$, $y$, $z$ not belonging to $S$, and $A(I) = (\{x, y\} \times (S \setminus I)) \cup (\{z\} \times I)$. Define a right $S$-action on $A(I)$ by $$(w, u)s =...
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Name of this commutative monoid over the divisors of a integer

Let $D_n$ be the set of divisors of $n$, and define a operation $\cdot$, such that $x,y\in D_n$, $x\cdot y=\gcd(xy,n)$. $(D_n,\cdot)$ is a commutative monoid. Is there a more well known name for such ...
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category of monoids has all coequalizers as follows.

The following is from Problem 13 of Chapter 3 in Awodey's Category Theory. 13-Show that the category of monoids has all coequalizers as follows. Given any pair of monoid homomorphisms $f, g: M \...
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Where can I learn about explicit homeomorphisms from $X=\Bbb Z[1/6]^+$ to $Y=\Bbb Z[1/2]\cap(0,1]$ in the 2-adic metric?

Where can I learn about explicit homeomorphisms from $X=\Bbb Z[1/6]^+$ to $Y=\Bbb Z[1/2]\cap(0,1]$ in the 2-adic metric? To be clear, I'm referring to $X$ the products of the positive dyadic and ...
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Prove that a monoid with property $ (a * a' = e) \lor (a' * a = e) $ is a group

Let $(G , *)$, $G \neq \emptyset$ be closed, associative, there is an identity element $ e $ in $ G $ and: $ (\forall a \in G)(\exists a' \in G) $ $ (a * a' = e) \lor (a' * a = e) $ Prove that G is a ...
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When are presheaves models of multi-sorted Lawvere theories?

As I understand it there is a correspondence between finitary monads and single-sorted (ordinary) Lawvere theories. My first question is, for a monoid $M$ in Set, when will it be the case that the ...
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Monoid actions - is this action inelegant?

I’m recently delving into abstract algebra, and I’ve attempted to devise a monoid action on the natural numbers. I think I must be missing something here—is there a better way to represent these same ...
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How is a monoidal category with one object a commutative monoid?

The nLab article for commutative monoid says: Just as a monoid can be seen as a category with one object, a commutative monoid can be seen as a bicategory with one object and one morphism (or ...
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Cannot become ring because distribution law does not hold

Commutative ring with unit is defined as $(R,+,\times)$, where $(R,+)$ is abelian group and $(R,\times)$ is commutative multiplicative monoid with $1$ and $+$ and $\times$ satisfies distributive law. ...
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set of all monoid homomorphisms $(\mathbb N_0, +) \to M$

Let $M$ be a monoid. Determine the set of all monoid homomorphisms $(\mathbb N_0, +) \to M$. I know that for a monoid homomorphism the following has to be true: $f(a\circ b) = f(a) \ast f(b)$ $f(e) = ...
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Show that if $f$ is a homomorphism then the set of invertible elements $M^\times$ is commutative

I have to show the following. Let $M$ be a monoid. If $M \to M$, $f: a \mapsto a^2$ is a homomorphism, then $M^\times$ is commutative. So, if $f$ is a homomorphism, then for all $a,b \in M^\times$ we ...
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Amalgamated coproduct of two copies of a partially ordered monoid $S$

The definition of the amalgamated coproduct $A(I)$ of two copies of a partially ordered monoid $S$ is as follows: Let $I$ be a right ideal of a pomonoid $S$, $x,y,z$ not belonging to $S$, and $$A(I)=(\...
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How to prove that in a semigroup, two $\mathcal{L}$ classes in a $\mathcal{D}$ class are incomparable by $\leq_{\mathcal{L}}$ relation? [closed]

Let $S$ be a semigroup. I would like to prove that two $L$ classes (namely $L_1$ and $L_2$) such that both are in a $D$ class (that is $L_1\subseteq D$ and $L_2 \subseteq D$) are incomparable by $\...
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Is the intersection of subsemigroups of a monoid $M$ that are also monoids, but with different identities from $M$, also a monoid?

As an example, consider the associative operation $\ast$, where $$ a \ast b = ab \mod{30}. $$ Note that $\ast$ is closed and associative on the sets $$S_1 = \{0, 1, 2, \dots , 28 , 29\}$$ $$S_2 = \{0, ...
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Does semigroup and monoid have to be closed under the binary operation?

As stated in the title, I am wondering whether semigroup and monoid have to be closed under the binary operation. The reason I am asking about this is that in wiki pages of semigroup and monoid, the ...
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Doubt in Location Lemma in Greens Relation Abstract Algebra!

I am unable to prove one part of rectangular lemma in green's relations. Let $S^1$ be a monoid. Then I need to prove that $m.m' \in D(m) \iff m.m'\in R(m) \cap L(m')$. How should I go about proving ...
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Finitely generated algebra and monoids

Let $R$ be a graded ring, with the grading induced by an abelian monoid $M$. I know that if $R$ is a finitely generated $R_0$-algebra, then $Supp(R)=\{w\in W\mid R_w\neq \emptyset\}$ is a finitely ...
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Does function composition with the set of all functions from $A$ to $B$ form a monoid?

Let $A = \{a, b, c\}$ and $B = \{a, b\}$. Denote the set of all functions $f : A \to B$ as $F$ and denote function composition in the typical way, i.e., $f \circ g = f(g(x))$. Is this a monoid? From ...
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The Transporter is a submonoid

I am reading Humprheys' Algebraic Groups, stuck at an apparently simple point In section 8.2 Actions of Algebraic groups(line 6, paragraph 1), The transporter is defined: Let $Y ,Z$ be subsets of $X$ (...
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How does $(eMe)m_1(eMe) = (eMe)m_2(eMe) \Rightarrow Mm_1M = Mm_2M$?

I am studying the book "Representation Theory of Finite Monoids" by Benjamin Steinberg, I've tried searching online for the solution but I don't know how to search for it so google shows no ...
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For which rings $R$ is $\operatorname{End}_{\mathbb Z}(R)$ isomorphic to $R$ (that is, the only endomorpisms are the multiplication maps)?

Let $(R,+,\cdot)$ be a commutative ring. It is a well-known fact that the ring of $R$-module endomorphisms of $R$, that is $(\operatorname{End}_R(R),+,\circ)$, is isomorphic to $(R,+,\cdot)$. Clearly, ...
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Affine semigroup generating a lattice

Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$). Assume that $S$ generates $N$ as a group. Is it true that it ...
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Artin Chapter 2, exercise 2.2: avoiding circularity

I'm worried that my answer to this exercise from Artin's book is trying too hard to balance avoiding verbosity with avoiding circularity. The exercise is: Let $S$ be a set with an associative law of ...
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Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits? [closed]

Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\...
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Weakening of GCD: maximal common divisor (MCD)

I'm dealing with integral domains and have come across the following weakening of the notion of a GCD (which I will write analogously as MCD): https://en.wikipedia.org/wiki/Maximal_common_divisor Let $...
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commutative law of monoid

can someone help me understand proof of this property? Let G be a commutative Monoid and $x_1,..,x_n \in G$. Let $\psi$ be a bijection of the set of intergers $\{1,2,3,...,n\}$ . Then $$ \prod^n_{v=1}...
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Is Set coclosed?

Background: $\mathbf{Set}$ is a (cartesian) closed monoidal category, so we have the natural tensor-hom adjunction $\text{Hom}(X\times Y,Z)\cong\text{Hom}(X,\text{Hom}(Y,Z))$ for sets $X,Y,Z$. ...
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Finding all small monoids with the help of GAP

In GAP there is the command AllSmallGroups(n) to construct all finite groups of order $n$ up to isomorphism. Question: Is there a similar method (or package) in ...
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Presentations of monoids

Is it true that every f.g. monoid is a quotient of a free one by some relations ? Given a monoid $M$ how can I construct the data in $$\Sigma^*/\sim~$$ ?
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Algebraic structures over groups and monoids [closed]

We define a vector space over a field and a module over a ring. But what algebraic structure is defined over a group? over a monoid? In case this can be done, could you give me examples?
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Let $(\mathbb{M},\cdot)$ be a finite monoid. Then there is some $n \leq |\mathbb{M}|!$ such that for every $k \in \mathbb{M}$ we have $k^n=k^{2n}$.

My attempt: Let $k \in \mathbb M$. Since $\mathbb M$ is finite, there exist integers $i,l > 0$ sucht that $k^i = k^{i+l}$. It follows that, for all $n \geq i$, we have $k^n = k^{n+l}$. Now if $n$ ...
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Modules over monoids vs algebra over monads

I read somewhere that the construction of algebras over monads is motivated by/ similar to the construction of modules over monoids, but I'm having difficulty seeing this. I see that a monad "...
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Why a monoidal category with only one object is a monoid in the category of monoids?

I'm reading Chapter 4 in Steve Awodey's Category Theory and it mentions A discrete monoidal category, that is, one with a discrete underlying category, is obviously just a regular monoid (in $Sets$), ...
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A tedious exercise about commutative monoids

From Lang's abstract algebra book: Let $G$ be commutative monoid, let $I$ be a set, and let $f:I\to G$ be a mapping such that $f(i)=e$ for all but finitely many $i$. Let $I_0$ denote the finite subset ...
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3 votes
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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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If multiplication by an element of a monoid is a bijection, is the element invertible?

Is it true that if multiplication on the right by a element of the monoid is a bijection, then the element is a invertible element? In other words, is it true that for the homomorphism $M \to \mathrm{...
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bijections between $\Bbb{N}^n$ and monoids morphism [closed]

Let $(L, \cdot)$ be a conmutative monoid with an identity $e$ and let $n \geq 1$ be a natural number. Show the following function is bijective. $F: Hom_{Monoid}(\mathbb{N}^n, L) \to L^n$ where $F(\...
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9 votes
1 answer
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Prefixes of a word multiplying to the identity in a free group

Let $A$ be a finite alphabet, and let $w \in (A \cup A^{-1})^\ast$ be a freely reduced word over the alphabet $A$ and formal inverse symbols $A^{-1}$. Suppose $w$ is non-empty. Can there ever be non-...
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1 vote
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Algorithm to check if a vector is in a finitely generated monoid

Suppose we are given a sequence of integral vectors $\alpha_1,\alpha_2,\dots,\alpha_m\in\mathbb{Z}^n$. For some $\beta\in\mathbb{Z}^n$, is there an effective algorithm to check if $$\beta\in\mathbb{N}\...
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-1 votes
1 answer
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Infintie monoids satisfying a relation [closed]

Let be $A$ a set. It is endowed with one internal composition law which is also associative, let's say $\cdot$. There are two elements $a_1, \: a_2\in A$, such that: $$a_1x^na_2=x,\forall\: x\in A$$ ...
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1 vote
1 answer
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Left inverse in monoid, left dual in monoidal category, and uniqueness

The notion of monoidal category is a categorification of the notion of monoid. If $M$ is a monoid, consider the monoidal category $Vec_M$ (of $M$-graded vector spaces over a field $\mathbb{k}$). If an ...
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3 votes
1 answer
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Inverse of ab in monoid implies a and b have inverses?

Let $a, b$ be elements in a monoid such that $ab$ has an inverse. Is it true that $a$ and $b$ have inverses? Prove this if true or give a counterexample if false. I believe this is false because let $...
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1 vote
2 answers
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Do sets of commuting elements having conjugates in a commutative submonoid have a single conjugating element?

It is known that any set of commuting diagonalizable matrices is simultaneously diagonalizable. So, it would be nice to ask the following generalization: Given a monoid $M$ (not necessarily a group), ...
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1 vote
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Bitstring algebra, monoid with concatenation and bitwise boolean operations

Given the free monoid on $\{0,1\}$ that is the set of all finite, possibly empty binary strings. $$\langle \rangle, \langle 0\rangle, \langle 1\rangle, \langle 01\rangle \in \{0,1\}^*$$ with the ...
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13 votes
2 answers
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Can all squares in a free group be made from squares in the free monoid?

Here's a question I thought of, that I don't know the answer to. Let $F_2$ be the free group on $\{a,b\}$, and $F_2^+$ be the subset where all the exponents are positive. For a set $S$, let $S^2=\{g^2:...
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4 votes
2 answers
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Is multiplication by a cancellative element a bijection between divisors?

Let $M$ be a commutative monoid and $m,x \in M$. Let $m^{-1}x$ be the set of elements $t$ such that $mt=x$ and suppose it not empty. Let finally $c$ be a cancellative element. Multiplication by $c$ ...
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