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

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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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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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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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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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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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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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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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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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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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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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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 , such that it 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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“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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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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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 ...
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Is there a name for those commutative monoids in which the divisibility order is antisymmetric?

Every commutative monoid $M$ is naturally equipped with its divisibility preorder, defined as follows. $$x \mid y \leftrightarrow \exists a(ax=y)$$ Is there a name for those commutative monoids such ...
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What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
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Proof of Paper, Scissor, Rock as Monoid Instance in Coq

So while learning Coq I did a simple example with the game paper, scissor, rock. I defined a data type. Inductive PSR : Set := paper | scissor | rock. And three ...
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Finitely generated ordered monoids and noetherian subsets

Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. Furnish $E$ with the order defined by "$x\leq y$ if and only if there exists $z\in E$ with $y=x+z$", so that ...
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Commutative monoids arising from categories with finite coproducts

If $\mathcal{C}$ is a category with finite coproducts, we may associate to it a commutative monoid $\mathcal{C}/\cong$ of isomorphism-classes of objects, with addition induced by the coproduct and ...
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Realizing the monoid $\mathbb{N}/(3=1)$ from a category with finite coproducts

If $\mathcal{C}$ is a category with finite coproducts (including an initial object $0$), then the set of isomorphism-classes of $\mathcal{C}$ becomes a commutative monoid with $0 := [0]$ and $[x] + ...
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Center of the categories $\mathbf{Grp}$ and $\mathbf{Ab}$.

This is Exercise II.5.8 from Mac Lane, Categories for the Working Mathematician. For the identity functor $I_C$ of any category, the natural transformations $\alpha:I_C\dot{\to}I_C$ form a ...
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Continuity of Multiplication by Fixed Element

This is likely a simple question that I'm just missing, but nothing immediately came to mind. When dealing with topological monoids, it is necessary to prove that the group operation is continuous. ...
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Every admissible ordering on a countably infinite monoid is induced by an isomorphism onto $(\Bbb{N}, \cdot)$.

Let $fg$ mean the functional composition of functions $f, g$. Let $M$ be a countably infinite, commutative, multiplicative monoid and let $f : M \to (\Bbb{N}_{\gt 0}, \cdot)$ be an isomorphism. ...
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Are these properties for a monoid enough for being the underlying monoid of an integral domain minus the zero?

If $R$ is an integral domain then $R-\{0\}$ equipped with the original multiplication can be recognized as a commutative and cancellative monoid. The inversible elements form a subgroup $R^*$ and it ...
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Finding a Monoid

I want to find the monoid $ colim_n \pi_0 (\mathcal{I} M_n(\mathbb{C})) $ which I know is isomorphic to $ \pi_0 (colim_n \mathcal{I} M_n(\mathbb{C})) $ where $ \mathcal{I}$ is the set of idempotents ...
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How many monoids of order three are there?

http://oeis.org/A058129 In the above link we can see the answer is 7. I have tried counting these and don't get 7. I am not sure what I am doing wrong so could someone go through counting these step ...
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Why is chosen for intersection instead of union?

Constructing a commutive monoid having idempotent elements (the underlying monoid of a Boolean ring) free over a set $X$, I arrive on a very natural way at monoid $M$ having the finite subsets of $X$ ...
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Semigroups and units

I am working my way though Basic Algebra 1. I am currently on chapter 1 , and more specifically I busy with the following exercise: "Let $S$ be a set a define a product in $S$ by $ab=b $. Show that ...