Questions tagged [monoid]
A monoid is an algebraic structure with a single associative binary operation and an identity element.
497 questions
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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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1answer
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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?
3
votes
2answers
54 views
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 $X,Y,Z\in\Xi$,...
0
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1answer
32 views
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 ...
1
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1answer
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 ...
1
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1answer
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{...
3
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0answers
26 views
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,+...
4
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1answer
103 views
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 ...
0
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0answers
15 views
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 ...
4
votes
3answers
64 views
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 ...
4
votes
0answers
48 views
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 ...
4
votes
1answer
130 views
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 ...
1
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1answer
29 views
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 ...
5
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0answers
85 views
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 ...
14
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0answers
127 views
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 ...
0
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1answer
28 views
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 ...
6
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0answers
67 views
+50
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 (...
1
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1answer
49 views
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 $...
1
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1answer
41 views
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 ...
1
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1answer
28 views
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 ...
1
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1answer
28 views
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 ...
2
votes
2answers
33 views
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 +...
0
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0answers
47 views
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....
1
vote
1answer
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 ...
3
votes
1answer
26 views
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 ...
0
votes
1answer
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⟩...
6
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3answers
142 views
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 $...
1
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1answer
49 views
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, $TX = \{[x_m,..,x_1] : m = 0,1,2,3..., x_i \in X\}$
$T$ ...
3
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0answers
60 views
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 ...
1
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1answer
87 views
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 ...
2
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1answer
53 views
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])...
1
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1answer
31 views
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 ...
0
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2answers
55 views
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 ...
1
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1answer
55 views
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 ...
0
votes
0answers
84 views
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
...
7
votes
1answer
134 views
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
...
2
votes
0answers
65 views
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 ...
1
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1answer
11 views
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 \...
0
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1answer
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 ...
1
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1answer
42 views
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 ...
1
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0answers
31 views
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{\...
0
votes
1answer
37 views
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 $F$ ...
0
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1answer
58 views
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 ...
6
votes
3answers
115 views
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 ...
1
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0answers
33 views
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 \...
1
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0answers
31 views
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 $\forall x,y \in S : axbayb = axyb$?
...
6
votes
1answer
101 views
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 ...
6
votes
3answers
291 views
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. ...
1
vote
1answer
26 views
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? ...
0
votes
1answer
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 ...
