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

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An easy example of a non-seminormal (commutative) monoid in generators and relations?

Let $M$ be a commutative monoid. We denote by $M^{Gr}$ its Grothendieck Group (i.e. group of fractions). We then have a morphism $q:M\rightarrow M^{Gr}$. We say that a commutative monoid $M$ is ...
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Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
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Where can I learn more about the “else” operation / “else monoid”?

(The set of natural numbers $\mathbb{N}$ starts at $0$ for me.) Let $X$ denote a set, and define $X_\bot = X \uplus \{\bot\}.$ Let $\mathbf{else}$ denote the binary operation on $X_\bot$ defined as ...
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Proving that the set of languages over an alphabet Σ is a monoid regarding concatenation

I'm practicing proofs and would like to prove that the set of languages over an alphabet $\Sigma$ is a monoid regarding concatenation by showing that the following statements are true: There is a ...
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32 views

submonoid of monoid.

Let $M = \{a,b,c\}^*$ be a free monoid. Let consider $M' = \{abc, abcba, baabc, baba\}^*$ Check, if $M'$ is a free submonoid of $M$ The solution is: $M'$ is not a free submonoid of $M$ beacuse: ...
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Reference for theorem on non-decreasing functions of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose $k\colon M\rightarrow N$ is a function such that $k(\epsilon)=\epsilon$ for all $a,b\in M$, there exists a unique $v\in N$ such ...
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Isomporphism two languages.

Let $A$ be an alphabet. Let $X,L \subset A^*$ $L$ is regular. Let $$X^{-1}L := \{ w \in A^* \mid \exists x \in X\ \ xw \in L \} $$ $$LX^{-1} := \{ w \in A^* \mid \exists x \in X\ \ wx \in L \} $$ ...
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74 views

monoids of injections and surjections

Let $X$ be an infinite set, and let $I(X)$ and $S(X)$ denote the monoids of injective and surjective maps from $X$ to itself, respectively. How do $I(X)$ and $S(X)$ relate algebraically? Is there ...
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220 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
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1answer
78 views

Epimorphisms of monoids

I know that in the category of groups, the epimorphisms are precisely the surjective homomorphisms. What about the category of monoids? One can easily see, that surjective homomorphisms are epic (even ...
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Neutral element in monoid.

Let $(G, *) $ be a monoid . Let $g \in G$ and $ g = g * g $. Can I assume, that $g$ must be neutral element? Why?
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Different interpretations of a monoid as a category

What is the relation between the categories $\mathbb{N}_0$ and $\mathbb{N}'_0$ as follows: Both objects and arrows of $\mathbb{N}'_0$ are the natural numbers and f is an arrow $f:a\to b$ iff $f+a=b$. ...
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1answer
89 views

Name for categorical product inside a monoid

If a monoid is a category with a single object, is there a "monoid-theoretical" concept that the categorical product translates to? As an analogue, in a poset the product translates to the notion of ...
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28 views

$G$ is a monoid such that cancellation laws hold in $G$

Let $G$ be a monoid such that cancellation laws hold in $G$ .Show that it is a group. I want to use the fact that if $G$ is non-empty set such that associativity holds in $G$ and the equations ...
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2answers
43 views

Relationship between submonoids and subgroups

I'm taking my first abstract Algebra course. During one of the last lessons, my teacher told us that If $G$ is a group and $M$ is a finite submonoid of $G$, then $M$ is a subgroup of $G$. For the ...
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40 views

A theory of radicals of integers?

It seems to me that radicals, natural numbers without power factors, generalize the concept of primes. You could ask after the nth radical and the number of radicals less than a specified number. But ...
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73 views

An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$

An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$ I can't do this one. How do I get started? It looks like it is saying there is only an inverse if ...
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Showing that the mapping $a\in M, a^3=1,x\mapsto ax^2a^2$ is an automorphism of $M$

I am confused I have noticed. Perhaps you can clear up my confusion, which I can't pinpoint the origin of. Let $M$ be a monoid and $a\in M$ such that $a^3=1$, show that the mapping $x\mapsto ...
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2answers
55 views

Is this a homomorphism? Example of a homomorphism where $f(1_S)\ne 1_T$

Let $f:S\to T$ be a surjective mapping of monoids that holds the homomorphic property: $f(xy)=f(x)f(y)$. I want to show that $f$ is a homomorphism and also find an example of a mapping that holds the ...
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1answer
26 views

A monoid can have only one zero

An element $z$ of a monoid $M$ is call a zero element if $xz=zx=z$ for all $x\in M$, can a monoid have more than one zero? I have put my attempt to prove this in the answer below.
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Is $(\Bbb N,\operatorname{LCM}(a,b))$ a monoid?

Is $(\Bbb N,\operatorname{LCM}(a,b))$ a monoid? Is it associative and does it have a neutral element? Refer to answer below for my attempt.
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Which of the following sets are monoids under addition?

Which of the following are monoids under addition? i) All integers ii) All even integers iii) All odd integers iv) all positive even integers v) all integers or halves of ...
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4answers
67 views

Is $S$ a monoid, or is $(S,*)$ a monoid?

If I have a set $S$ with operation $*$ as a monoid. Would I say I have a monoid $S$ with the binary operation $*$ or would I say I have a monoid $(S,*)$ where the binary operation $*$ does ...
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When can a given element in a monoid be decomposed into the product of a given element and another element?

Let me begin with the particular monoid that I was origionally interested in, which was the set of real valued functions of a single real variable with the composition operation. My question was this, ...
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102 views

Is there a name for property $n+k=m+k\implies n=m$?

Monoid of natural numbers with addition have such property, that for any $n,m, k \in \mathbb{N}$ if $n+k=m+k$ then $n=m$. Does this property have some name in English?
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If $X$ is an Abelian group, then $\ker_X : \mathrm{Cong}(X) \rightarrow \mathrm{Sub}(X)$ is a bijection. Is there a partial converse?

(All monoids are written additively in this question, even the non-commutative ones.) Given a monoid $X$, write $\mathrm{Sub}(X)$ for the lattice of submonoids of $X$, and write $\mathrm{Cong}(X)$ ...
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Does there exist a surjective homomorphism between every pair of monoids?

Say we have two monoids $N,M$ and w.l.o.g. assume that $|N| \geq |M|$. Does there exist a surjective homomorphism $\varphi : N \to M$? Context In the category of sets the answer would be yes: there ...
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1answer
59 views

Group-like structures over the integers and functions on them

The integers with addition build a group $\langle \mathbb{Z},+,0\rangle$. The functions $\operatorname{succ}:\mathbb{Z} \rightarrow \mathbb{Z}$, $\operatorname{pred}:\mathbb{Z} \rightarrow ...
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What is the name of the algebraic structure constructed with an abelian monoid and a field?

Take a vector space built from an abelian group $(V,+)$ whose elements are the vectors, a field $K$ whose elements are the scalars, and there is an operation (multiplication by scalars) that ...
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63 views

Prove that in a finite monoid each element is invertible

Let $(M,\circ)$ be a finite monoid. Suppose the identity element $e\in M$ is the only idempotent element. Then prove that each element in $M$ has inverse. How can I prove this?
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Subsets of a monoid closed under left-multiplication by elements of a submonoid

Let $M, T$ be monoids (or, semigroups) with $M \subset T$. Then we can consider subsets $S$ of $T$ that are closed under left-multiplication by something in $M$, i.e. $$ a \in S, m \in M \implies ma ...
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59 views

References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
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If two elements a,b have a gcd, do then a*t,b*t also have a gcd?

Let $M$ be a commutative cancellative monoid. For elements $a,b \in M$ a gcd of $a,b$ is an element $\mathrm{gcd}(a,b)$ with the (universal) property $\forall c \in M (c |\mathrm{gcd}(a,b) ...
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67 views

Building a commutative monoid in which every cancellative element is invertible

I think that for every commutative monoid $M$, there should be a corresponding commutative monoid $F(M)$ satisfying the following implication. $$\forall a(\forall x,y(ax=ay \rightarrow x=y) ...
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Does such a finite monoid exist?

Does there exist a finite monoid $M$ such that for some $x \in M,$ the following hold? $x$ cancels on both the left and the right: $$\frac{ax=bx}{a=b}\qquad \frac{xa=xb}{a=b}$$ $x$ has no two-sided ...
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Trying to understand significance of monoid as a one object category

So I generally understand the idea of a monoid from set theory, but I'm trying to understand better the mapping to category theory. http://en.wikipedia.org/wiki/Monoid#Relation_to_category_theory I ...
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Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...
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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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Operations on power set

Which kind of operations on a power set leads to a group/monoid? Known to me are: - intersection - union - symmetric difference - complex product of a group Ate there some more examples?
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Are the identities $\{(xy)^p = x^p y^p : p \mbox{ is prime}\}$ logically independent?

For each positive integer $n$, let $\eta_n$ denote the following identity in the language of monoids. $$(xy)^n = x^n y^n$$ For example, $\eta_2$ is the identity $xyxy = xxyy.$ Question. Is it ...
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Injectivity of Natural Homomorphism to Groupification

This is a continuation of my own question some time ago. Suppose $M$ is a monoid and $G$ is the groupification of $M$. (I figure groupification of $M$ is a better term than Grothendieck group of ...
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Example of a semigroup with unique idempotent which is not a monoid

I am searching for an example of a semigroup, with unique idempotent element, that is not a monoid. Please help.
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Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
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group-like structure texts.

I was reading Dummit and Foote to be ready for my group theory text, but my teacher seems to be paying special attention to things with less structure than groups, for example monoids, semigroups, and ...
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Showing uniqueness of inverse element of an element of a monoid

Question- If $\langle A,*\rangle$ is a semigroup with identity, prove that every element a belonging to $A$ has at most one inverse. Proof- Let the identity be $e$. Let us assume that $b_1$, $b_2$ ...
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2answers
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“The regular languages over $A$ are the homomorphic pre-images in $A^*$ of subsets of finite monoids.”

I'm trying to understand the statement: The regular languages over $A$ are the homomorphic pre-images in $A^∗$ of subsets of finite monoids. which appears in the Wikipedia article on free ...
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2answers
65 views

May a monoid have two disjoint submonoids?

I'm asking this question inspired by the similar question about group and its subgroups. I tried to modify the proof presented there to work for monoids but I failed. I'm also not able to find any ...
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70 views

If $G,H$ are monoids and not groups, prove that $f(e_G)=e_H$ may be wrong.

Problem: If $f:G\longrightarrow H$ is a homomorphism of groups, then $f(e_G)=e_H$ and $f(a^{-1})=f(a)^{-1}, \forall a\in G$. Show by example that the first conclusion may be false if $G, H$ are ...
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What are some examples of non-commutative monoids that are both idempotent and self-distributive (on both sides)?

In the presence of the axioms for a commutative monoid, idempotency is equivalent to self-distributivity. Proof. Suppose a commutative monoid is idempotent. Then: $$x(yz) = xxyz = (xy)(xz)$$ On the ...
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Monoidal product is coproduct in category of commutative monoids

If $V$ is a symmetric monoidal category, the category $\text{CMon}(V)$ of commutative monoids in $V$ has binary coproducts given by $\otimes$, the monoidal product of $V$. See for example Johnstone’s ...