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Questions tagged [monoid]

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

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About monoid homomorphism

Let's say I have two sets $A$ and $B$, which are power sets of natural numbers less than or equal to $1$ and $2$ respectively. So $A$ = {$\emptyset$, {$0$}, {$1$}, {$0,1$}} and B = {$\emptyset$, {$0$},...
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Is (topology, union, empty set) with basis as generator a monoid?

Set - $\tau $ Generator - basis of $\tau $ Operator - $\cup $ Unit - empty set Is it a monoid?
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Burnside convolution

Let $G$ be a group. Say that an orbit is a nonempty transitive $G$-set. Let $\Xi$ be a set of finite orbits such that each finite orbit is isomorphic to exactly one element of $\Xi$. If $X,Y,Z\in\Xi$,...
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Show that $\equiv$ is a congruence on $M\times S$

I'm sorry if a similar question has been posted before, but I was unable to find one based on my searches. This is an extra practice problem for a number theory class. I've been trying to prove this ...
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13 views

Cyclic monoid isomorphism

Prove that any cyclic monoid is either isomorphic to (N, +) or is isomorphic to a monoid of the form of a finite cyclic monoid of some size. I understand that this is saying that a cyclic monoid is ...
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23 views

Is there a sum-like operator over subsets of 1 or 2 elements over $\mathbb{Z}$

Let be $A(\mathbb{Z})$ the set of subsets of 1 or 2 elements, like: $\{ 1 \}, \{ 1, 2 \}$. I would like to know if we could prove there is no map $+ : A(\mathbb{Z}) \times A(\mathbb{Z}) \to A(\mathbb{...
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What is the Krull dimension of the Burnside ring of $\mathbb N$?

A contravariant functor $F$ from monoids to commutative rings was defined there. Question. What is the Krull dimension of $F(\mathbb N)$? (Here $\mathbb N$ denotes the additive monoid $(\mathbb N,+...
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From monoids to commutative rings

We shall first define a functor $$ F:\mathsf{Mon}^{\text{op}}\to\mathsf{CRing}, $$ where $\mathsf{Mon}^{\text{op}}$ is the category opposite to the category of monoids and $\mathsf{CRing}$ is the ...
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Is every summation structure a complete monoid?

A summation structure is a pair $(S,\sum)$ with $S$ a set and $\sum:S^I\to S$ is a map such that For $I=\{i\}, \sum \alpha=\alpha i$ For $\eta:I\to A$ and $\beta:A\to S$ with $\beta a=\sum ...
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3answers
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Group-subsets of monoids with different identities

A subgroup must have the same identity as its containing group, but this fact requires inverses. I'm interested in subsets of monoids, which are groups in their own right, but vary greatly from the ...
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supremum of additive functions is additive

I need some help for one equality in the following proposition. It was a hint to conclude that $\sup\{f(\cdot):f\in\Phi\}$ is additive. I highlighted it blue. Ultimately I am interested in proving the ...
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Why are monoids not treated in most algebra courses?

I’ve taken a look at a number of introductory books on abstract algebra. They all treat groups, rings, and fields, and many of them treat galois theory, linear algebra, algebras over fields. But none ...
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Finite generation of cones in $\mathbf{Z}^n$

Let $X$ be a finite free $\mathbf{Z}$-module of rank $n$ and let $X_+ \subset X$ be a cone i.e. a subset of $X$ such that for any $x,y \in X_+$ and $a,b \in \mathbf{N}$ we have $ax+by \in X_+$. Is ...
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A counterexample to show that the following set does not form a semigroup

Let $A = $ { $f:\mathbb Z$ $\to \mathbb Z$| the cardinality of set {$x \in \mathbb Z$ | $f(x) = x$} is finite}. I have to prove or disprove that the set $A$ forms a semigroup/monoid under function ...
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Power of subset of finite group is a subgroup. [duplicate]

Let $G$ be a finite group and $S$ a nonempty subset of $G$. I want to prove (or disprove) that $S^{|G|}$ (that is products of length $|G|$ of elements of $S$) is a subgroup. My work so far : Since ...
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Showing that the following set of functions forms a monoid on a base set X

Let $M$ be a monoid of functions on a base set X. Define $S$ = {$f$: $X$ $\to$ $X$ | $m$ $\circ$ $f$ $=$ $id_X$ for some $m$ $\in$ $M$}. Show that $S$ is a monoid of functions on $X$. Attempt: I am ...
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Bounding the number of certain (translation) subsets in $\mathbb N \times \mathbb N$ with respect to given subsets

Let $\mathbb N = \{0,1,2,\ldots\}$ be the monoid of natural numbers with zero. Suppose $S \subseteq \mathbb N \times \mathbb N$ be some subset such that the number of sets of the form $\{ (i,j) \mid (...
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What properties of $R$ does the monoid ring $R[M]$ inherit?

It's known that polynomial rings $R[x_1,...,x_n]$ inherit some properties of a base ring $R$. For example, (due to the wikipedia article on polynomial rings) if $R$ is an integral domain, then so is $...
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About identity of a monoid

This question is from Jonathan D. H. Smith's Introduction to Abstract Algebra book. The question basically says that for an infinite set $X$ a function $f : X \to X$ is called almost identical if ...
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How to check if the identity function lies in a given set of functions to verify monoids?

Let's say I have a set $A$ and $a$ $\in$ $A$. Suppose I have a set $N$ = {f $\in$ $A^A$| $f(a) = a$}. Now, I want to check if $N$ forms a monoid of functions on $A$. Now, by verifying associativity ...
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Question on some specific property of ordered semigroup

Let $\langle S, \cdot, \leq \rangle$ be an ordered semigroup (or monoid). Suppose, we have some element $a \in S$, such that for each $b \in S$, $a \cdot b \leq a \cdot b \cdot a$ and $b \cdot a \leq ...
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Series of binary operations in $(\mathbb{N}, *)$

Consider monoid $(\mathbb{N},*)$, where operation $*$ is defined as $x*y = xy + x + y$. What is the result of $1*2*3*\text{...}*25 \text{ (mod 29)}$ ? $xy + x + y$ can be written as $(x + 1) y +...
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How to think about this monoid?

Let $P$ be a commutative monoid. Consider the $P$-monoid $P^{\frac{1}{n}}$ which is any monoid isomorphic to the monoid $n:P \to P$ ( $x \to x^n$). I'm using the multiplicative notation for my monoids....
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31 views

Can every monoid be turned into a ring?

It’s well-known that $\mathbb{Q} / \mathbb{Z}$ is an example of an abelian group which is not isomorphic to the additive group of any ring. But my question is, does there exist a monoid which is not ...
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Are simple commutative monoids monogeneous?

Let $M$ be a simple commutative monoid. Is there a surjective monoid morphism $\mathbb N\to M$? If non-monogeneous simple commutative monoids do exist, what's known about them? Edit. The ...
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27 views

Submonoid generated by a subset

Definition: If $S$ is a subset of monoid $[M;∗]$, the submonoid generated by $S$, $⟨S⟩$, is defined by: The identity of M belongs to $⟨S⟩$; and $a ∈ S ⇒ a ∈ ⟨S⟩$ $a, b ∈ ⟨S⟩ ⇒ a ∗ b ∈ ⟨S⟩...
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How to classify $\mathbb N^2$-orbits?

I'm sure the answer to this question is well-known, but I don't know how to search for it. Neither do I know how to tag the question. Feel free to retag it! Or to mark it as a duplicate! Say that an $...
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The category of $T$ algebras on Set is equivalent to the category of monoids

Let Set denote the category of sets. Let $T:$ Set $\to$ Set be the functor that sends a set $X$ to the set of finite words on $X$. That is, $TX = \{[x_m,..,x_1] : m = 0,1,2,3..., x_i \in X\}$ $T$ ...
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How many endomorphisms $(\mathbb{N},\times)$ has? Like object in Mon category

I was told that there is only one morphism in $Mon$ category for this object $(\mathbb{N},\times)$. But why? I think that we can write every natural number as the product of a certain set of prime ...
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Question about construction of The Grothendieck group.

In the Algebra by Serge Lang, he constructed a Grothendieck group of commutative monoid $M$, namely $K(M)$:(page 39-40) $M$ is a commutative monoid. Let $F_{ab} (M)$ be the free abelian group ...
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Are multiplicative monoids of different rings isomorphic?

I have some algebraic problem and hope some of you can help me! The problem is about rings and their multiplicative monoids (semi-groups with neutral element $e$). So $M(\mathbb{Z}), M(\mathbb{Z}_2[x])...
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Neutral element for a Cantor Set.

On Wiki I found the following statement: $T_L$ and $T_R$ together with function composition forms a monoid. I am able to prove the associativity of the composition operation, but what will be a ...
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Underlying set of the free monoid, does it contain the empty string?

In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element. Is the empty string an element of the underlying set of the free monoid? In ...
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Question regarding monoids.

Let $(X,\circ,e)$ be a monoid, for set $X$, binary operation $\circ$, and identity element $e$. Suppose $$ A_1 \circ A_2 \circ \ldots \circ A_n = B_1 \circ B_2 \circ \ldots \circ B_n = C, $$ where ...
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Elements of the Monoid in the category of endofunctors

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
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Monoid in the category of endofunctors and Monoid as a category with one object

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
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Name for mathematical structure that generalizes function composition and application

I was wondering what the name is for a mathematical structure equipped with two operations, composition ($\circ$ or jutxtaposition) and application ($\cdot [ \cdot ]$) where the two operations satisfy ...
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Injective function between a monoid and a function from same monoid to monoid

Suppose we have $M$ a monoid. Define $E(M) = \{\alpha : M\rightarrow M : \alpha(xy)=\alpha(x) \cdot y \}$ If $a \in M$, define $\alpha_{a}: M \rightarrow M$ by \begin{align*} \alpha_{a}(x)=ax \...
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86 views

Binary operation with complex number

Let us consider the set of complex numbers and the binary operation $\circ$ defined by $z_a\circ z_b=|z_a|e^{\Theta(z_b)}$, where $\Theta(z_b)$ is the argument of the complex number $z_b$. Explain ...
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1answer
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Complex function with binary operation

I'm working on a question on complex function with binary operation, it has two parts in the question and I got stuck on part b. Part a: represent the $∘$ operation in a graphical way. I have drawn ...
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Replacing $H$-space multiplication with a fibration

Let $X$ be an $H$-space with multiplication $\mu: X \times X \to X$. Does there exist a space $\overset{\sim}{X}$ homotopy equivalent to $X$ such that the induced map $\overset{\sim}{\mu}: \overset{\...
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Consider the set F under the operation of composition of functions ◦.

Let $C = \{z \in \mathbb C \mid |z| = 1\}.$ Let $f_\theta : \mathbb C \to \mathbb C$ be given by $f_\theta (z) = e^{i\theta z}$. Let $F = \{f_\theta | \theta \in \mathbb R\}$. Consider the set $F$ ...
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Classify monoids that are generated by one element.

Algebra by Michael Artin Exer 2.M.4 M.4. A semigroup S is a set with an associative law of composition and with an identity. Elements are not required to have inverses, and the Cancellation Law ...
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Is a monoid commutative if $(ab)^2=a^2b^2$?

Let M be a monoid. Suppose that: $(ab)^2=a^2b^2$ for any elements a,b in M. Is M commutative? The result is obviously true for groups, but I can't find a counterexample for monoids. And without ...
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A proof for condition under which a monoid must also be a group: is it correct?

Let $(M, \cdot)$ be a monoid, where $|M| = k$, for some $k > 1, k \in \mathbb{N}$. The elements of this monoid have a special property: for any $a, b, x \in M$: $$ a \cdot x = b \cdot x \...
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Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$?

Based on J.-E. Pin's answer here, I'd like to know the following: Question. Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$? ...
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Terminology for a “subgroup” that has a different identity element.

Let $M$ denote a monoid. Then to refer to submonoids of $M$ that just happens to be a group, I think the phrase "subgroup of $M$" is okay, as it's unlikely to cause confusion as long as you instruct ...
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3answers
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In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column?

In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column? It's obviously not the Closure axiom, and after some inspection, I believe the Inverses axiom does fail. ...
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Is the monoid ring of a Noetherian monoid Noetherian?

Let $M$ be a monoid. Suppose that $M$ is left Noetherian, i.e. that every increasing chain of left ideals in $M$ stabilizes. Then is the monoid ring $\mathbb{C}[M]$ necessarily a left Noetherian ring? ...
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51 views

Monoid in general dynamic system definition

I am a newbie in this field but what difference does taking monoid or group in the following definition of dynamic system make? A tuple \begin{equation} (T,M,\phi) \end{equation} is called dynamic ...