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

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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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25 views

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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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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31 views

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 ...
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Are these equivalent definitions of the Grothendieck group construction?

I'm learning K-theory and I'm slightly confused. Given an abelian monoid $M$, we can construct the Grothendieck group $$ G(M) = (M \times M)/\sim$$ and I've seen two definitions for the equivalence ...
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Monoid Ring of a Commutative Cancellative Ordered Monoid

Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by $$ m \preceq m' \text{ if and only if there ...
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Concluding about the relation of two monoids to a third a relation between the two monoids

Let $N, S, T$ be monoids and let $\varphi : N \to S$ and $\varphi' : N \to T$ be surjective homomorphisms such that for each $P \subseteq S$ there exists some $Q \subseteq T$ such that $$ ...
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Monoid and category

Let $M$ be a monoid. Let $a, b$ be elements of $M$. We denote $\operatorname{Hom}(a, b) = \{s\ |\ sb = a\}$. Then we get a category whose set of objects is $M$. We denote this category by $C(M)$. Let ...
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Recovering Ordered Monoid Operation from the Order

I have a partially ordered set $(X, \preceq)$ with the following properties: $X$ has a minimum. I'll name it $1$. For every $x \in X$, the principal filter ${\uparrow} x$ is order isomorphic to $X$. ...
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Translation help (French): “Monoïde cycliste”

I am reading this paper, and in section 4, they define the "monoïde cycliste" to be the quotient of the free monoid on two generators, $\{\alpha,\beta\}^*$, by the relations $\alpha\alpha\beta \sim ...
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Inclusion-Exclusion Proof without inverses

If $f$ is additive then it satisfies the inclusion-exclusion principle $$ f(\bigcup A) = \sum_{B \subset A} f(\bigcap B) \cdot (-1)^{|B|+1} $$ See ...
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yes, why does a negative times a negative make a positive? [duplicate]

for a while I have been interested in the details of the construction of the integers from the natural numbers. credit to the software, for as soon as I began writing this, for it drew my attention ...
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Formal definition of string-to-term substitution

Forgive me if this question seems elementary. Let L be a nonempty alphabet. We have an intuitive notion of substituting words for letters in a word. For example, substituting 'cd' for 'b' in the word ...
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Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$ f(g(m)) = m $$ I want to understand what all of the ...
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Show that $M$ is a submonoid of the group of $2\times 2$ matrices of integers mod $13$.

Define $$M=\left\{\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\in(\Bbb Z/13)_{22}:a_{12}=0\right\}\;.$$ (This is the set of $2\times 2$ matrices with integers mod $13$ with the ...
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partitions are the elements of the free abelian monoid on $\aleph_0$ generators

although the following insight may not be particularly profound, it is always of interest when we see a canonical isomorphism between objects of apparently different types. it is a commonplace that ...
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Submonoids of $\mathbb{N}^k$

Do you know if all submonoids of $\mathbb{N}^k$ are finitely generated? If not, can you give me a counter-example? EDIT : I mean $\mathbb{N}^k$ as a submonoid of $(\mathbb{Z}^k,+)$. I already know ...
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Boolean endomorphisms vs endofunctions on finite sets

I stumbled upon a funny fact: Let $\mathbf{Bool} = \{0, 1 \}$. For all functions $f: \mathbf{Bool} \to \mathbf{Bool}$ it is the case that $f^3 = f$. This got me excited and I was wondering whether ...
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On the powerset of a monoid, does the second operation play nice with the first? Indeed, is it even useful?

Let $X$ denote a monoid. Then we can make $Y = \mathcal{P}(X)$ into a monoid, too. Define $$AB = \{ab \mid a \in A, b \in B\}$$ for all $A,B \in Y.$ We see immediately that $1$ (shorthand for $\{1\}$) ...
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Localization $Q-F$ of an affine monoid $Q$ is an affine monoid

Let $F$ be a face of an affine monoid $Q$. I have to show that $Q-F=\lbrace q-f \mid q\in Q, f\in F\rbrace$ is again an affine monoid. I have no clue about this, any help would be appreciated.
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Terminology regarding elements of monoids

In what follows, the symbols $a,b$ and $n$ implicitly range over $\mathbb{N} = \{0,1,2,\cdots\}.$ Are there names for the following properties that an element $x$ in a monoid may or may not possess? ...
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A finite Monoid $M$ is a group if and only if it has only one idempotent element

Suppose that $(M,*)$ is a finite Monoid. Prove that $M$ is a group if and only if there is only a single idempotent element in $M$, namely $e$. One direction is obvious, because if $M$ is a group ...
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If every element in a finite monoid $(M, \star)$ is uniquely-regular then $M$ is a group.

Suppose that $(M, \star)$ is a finite Monoid and for every $m \in M$ there exists a unique element $n \in M$ such that $m\star n \star m=m$ or in other words every element in $M$ is (uniquely!) ...
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Chinese remainder theorem and order

How do I find out which elements of the monoid (Z/161Z, *, 1) are not invertable? I'm trying to find the group of units but I can't really grasp which elements are invertible and which aren't.
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Examples of groups of mapping over a set which is not a subset of the symmetric group.

We can find examples of sets $X$ for which there exists a group $G$ (with |G| > 1) under function composition which is a subset of $X^X$ but not a subset of the group $Sym(X)$. The catch is that the ...
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Does there exist a reversible monoid that fails to be Dedekind-finite?

Call a ring with unity reversible iff $xy = 0$ implies $yx = 0$. Dedekind-finite iff $xy = 1$ implies $yx = 1$. It is proved here that every reversible ring is Dedekind-finite. Now clearly, the ...
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Showing MON and CAT are equivalent categories

I am struggling with an equivalence of categories. Let $\mathbf{Mon}$ be the category of monoids, and let $\mathbf{Cat}$ be the category whose objects are all categories with exactly one object. The ...
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Monoids as categories; does this construction have a name?

We can view a monoid $M$ as a category with a single object. However, there is another way to make $M$ into a category. Take the elements of $M$ as objects, and define $\mathrm{Hom}(x,y)$ to be set of ...
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Congruence induced by a subset.

Consider the additive monoid of natural numbers and calculate the congruence generated by $\{(2,3)\}$. I know the answer is that the congruence has 3 classes which are $\{0\}$, $\{1\}$, and the rest. ...
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homomorphism problem

The monoids $(S,\min)$ and $(T,\min)$, where $S = \{3,4,5,6\}$ and $T = \{1,2,3,4,5,6\}$ and $\min$ is the minimum function of two integers. Let the function $f:S \to T$ be defined by $f(x) = x-1$; ...
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The intersection of all ideals of a monoid

While reading about sandpile groups, I read the (rather important) assertion that the intersection of all ideals of a commutative monoid is a group , but was unable to find a proof. Unless the monoid ...
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Is it possible that $(ab)^{-1}$ is defined although $a^{-1},b^{-1}$ are not?

I wish to enquire about the properties of units in abstract algebra. In a ring $R$, a unit $u$ is an invertible element. Let $u=ab$. Is it possible that $a$ and $b$ are not units? Is it possible ...
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A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
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Basic examples of monoids?

What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they ...