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

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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 ...
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Do normal group endomorphisms form a normal submonoid?

What it says on the tin. A group endomorphism $v\colon G\to G$ is called normal if $v(aba^{-1})=av(b)a^{-1}$ for all $a,b\in G$. Equivalently, the map $g\mapsto v(g^{-1})g$ is a group homomorphism. ...
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Why are the algebras of the associative operad unital?

According to the n-lab page: The associative operad Assoc is an operad which is generated by a binary operation $\Theta$ satisfying $$\Theta\circ(\Theta,1)=\Theta\circ(1,\Theta)$$ It then ...
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a characterization for subgroups of cancellative monoids.

Suppose $S$ is a cancellative monoid and $A\subseteq S$ and $$\{(x,y)\in S ^2\mid y\in Ax\}, ~ \{(x,y)\in S ^2\mid y\in xA\}$$ are equivalence relations on $S$. Is $A$ a group?
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Grothendieck Group for General Monoids

Is there an analog to Grothendieck group for general monoids? I imagine such thing could be constructed but I have not found this construction in standard text so I am not sure if it makes sense. (Or ...
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Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$. They are uniformisable and hence are ...
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Ring Embeds in Monoid Ring

Let $(S,+)$ be a nontrivial commutative monoid and $R$ be a ring. Prove that $R$ embeds in $R[X;S]$ via $a \to aX^0$ I'm not exactly sure how to approach this... I think I may need to use the fact ...
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Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
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What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
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Define $ker(f)$ for $f:(\mathbb{Z},+)\to (\mathbb{Z}/_{12},+)$

I need to define the $ker(f)$ for the following homomorphism $f:(\mathbb{Z},+)\to (\mathbb{Z}/_{12},+)$ Where $f(x) = [x]_{12}$ If I understood that correctly then the $ker(f) = \{a \in \mathbb{Z} ...
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Monoid on ordered partitions of a natural number

Fix a natural number $n$, and let $O_n$ be the set of ordered partitions of $n$. For example $O_3=\{1+1+1,1+2,2+1,3\}$ which can also be written as $\{1|2|3,1|23,12|3,123\}$. We can define two ...
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Is it true to say that every tensor is an element of a monoid?

If we consider that: by definition, a tensor is an element of the tensor product of two algebraic structures the most abstract algrebraic structure on which the tensor product is defined are ...
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Relations between monoids and modules?

What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a ...
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positive integer polynomial under the usual polynomial multiplication

consider the set of polynomials with positive integer coefficients together with the operation, usual multiplication of polynomials. now my first question is does this set together with the ...
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Findout if $\min(x,y)$ and $\max(x,y)$ are semigroup, monoid or group

I need to find out whether $A \oplus B := \max(A,B)$ or $A \ominus B := \min(A,B)$ form a semigroup, monoid or group for $\oplus : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and $\ominus : ...
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In a dagger category, what do we call an arrow $f$ such that $f \circ f^\dagger \circ f = f$?

In a dagger category, what do we call an arrow $f$ satisfying $f \circ f^\dagger \circ f = f$? In $\mathrm{Rel}$ (and, more generally, an allegory) this is straightforwardly equivalent to $f \circ ...
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Cardinality of a Monoid and Constant Functions

Let $X$ be a set. Show that $((M(X),\circ)$ has an absorbing element iff $|X|\leq 1$ iff $M(X)$ is commutative. In this problem $((M(X),\circ)$ is a monoid and M(X) is the set of all maps from X to ...
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Monoid Definition

A monoid is $(M,*,e)$. I am having trouble understanding what the 'multiplication' operation represents. Is it an operation for an element which returns another element or an operation which takes ...
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On identity elements in monoids

In the definition of a monoid firstly we should have associativity. What I wonder about is the definition of the identity element; $\exists x \forall y\;\; x.y=y.x=y $ Which structure do we get if ...
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Are Monoids a category inside a category?

Looking at the definition of Monoids, it looks like they are an object inside a category with one object. I have also noticed that they have operations like composition and identity which must be ...
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Generalising cover maps from monoids to semigroups

Let $T,S$ be monoids. A partial surjective mapping $\psi : T \to S$ is called a cover map if for each $s \in S$ there exists some $\hat{s} \in T$ called a cover of $s$ such that for each $t \in ...
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Which of these constructions are left adjoints?

A groupoid can be regarded as a small category in which every arrow is an isomorphism A monoid can be regarded as a small category with only a single object A preorder can be regarded as a small ...