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

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Can the order on an ordered, cancellative monoid be extended to its Grothendieck group?

Suppose we have an ordered, cancellative monoid and we wish to apply the Grothendieck group construction to it. Can the total order be extended to the larger group? Example: consider the ordered ...
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Monoid filtration

I lately been introduced to monoid filtrations and I have a couple of questions. Let $(\mathfrak{M},\star,1_\mathfrak{M})$ be a monoid with total order, $(A,+)$ the additive subgroup and ...
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Is $\langle\mathbb Q^+, *\rangle$ a monoid?

Q: Given the set of positive rational numbers $\mathbb Q^+$, the operation is multiplication$~*$. Is $\left<\mathbb Q^+, *\right>$ a monoid? My answer is: $ \forall x, y, z \in \mathbb Q^+$, ...
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What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $S \leftarrow ...
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Is there a synthetic definition of the $0$-Hecke monoid of $S_n$?

Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$-th symmetric group. The $0$-Hecke monoid $H_0\left(S_n\right)$ is defined to be the monoid given by generators $t_1, t_2, ...
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Are all elements of groups Garside elements?

Here in the definition of Garside Element, it seems that if the monoid here is a group, then all its elements are Garside elements simply because all elements are both left and right divisors of any ...
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localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$

Let $X$ be an $H$-space. Let $F$ be a field. Then $H_*(X;F)$ is a Hopf algebra over $F$. According to group-like elements in the Hopf algebra of the homology of H-spaces, $$ \pi_0(X)=\{g\in ...
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Computing the shortest encoding for a transformation

Here goes... Let $n = 2^p$ for some $p \in \mathbb{N}$. Let $\mathcal{T}_n$ denote the set of all transformations on $\mathbb{Z}_n$ (viz. the transformation monoid). Pick $C$ to be an order ...
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configuration-spaces and iterated loop-spaces

In the paper Configuration-Spaces and Iterated Loop-Spaces. Graeme Segal, page 213-214, it is obtained that the labelled configuration space $C_n$ is homotopy equivalent to a topological monoid ...
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Monoids in Category Theory

I don't have a strong math background (engineering math) so I am at a bit of a disadvantage here but I have been trying to learn the broad strokes of Category Theory to help get a fuller picture of ...
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Groups of the from $gMg$ in a monoid where $g$ is an idempotent

Let $(M, \cdot)$ be a finite monoid with identity $e$. It is easy to see that $gMg = \{ gxg : x \in M \}$ forms a monoid with identity $geg = g$ if $g$ is an idempotent. If $gMg$ contains no ...
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Associativity of product law in $R^S$ ($R$ ring, $S$ a monoid with condition)

In Proof of associativity of polynomials product (infinite variables), I ask a question about polynomials and assume it was linked to a question of total algebra. I explicitely ask this question here ...
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when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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Monoid as a single object category

I'm struggling with comprehending what monoids are in terms of category theory. In examples they view integer numbers as a monoid. I think I get the set theoretic definition. We have a set and a ...
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Functions in the definition of Universal Mapping Property of a free monoid

In Awodey's Category Theory (http://www.amazon.com/dp/0199587361/?tag=stackoverfl08-20) p.19, what is $|\bar{f}|$? What is its relation to $\bar{f}$, which is in the existence statement? Is it ...
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Modules over a monoid: trouble with the definition.

I'm having trouble with a definition. We're working in the category of monoids. Take $A\in \mathfrak{Mon}$ and define a module over $A$ to be a set $M$ with an action: $A\times M \rightarrow M$ such ...
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Determine if given set is a free submonoid.

Determine such A, that $A^*$ is a free submonoid of $\{a,b,c,d,e,f,g\}^*$. A) $A = \{ ae, b,c,de\}$ B) $A = \{ ade, ddbee,dfc,dgd\}$ C) $A = \{ a, ab,bc,c\}$ D) $A = \{ ab, ba ,ca\}$ E) $A = \{ ...
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Homomorphism. Equation.

Let $h$ be a homomorphism monoid $M =\{0,1\}^*$ $h: M \to M, h(0) = 1, h(1) = 010$ And it is true: $h^3(1^+) = (h(1)h(1)h(1) )^+ $ I don't understand this equation.
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homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...
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configuration space model for classifying space of monoid

Let $M$ be a monoid and $BM$ be its classifying space. There is a model for $BM$ based on labelled configuration spaces of the line $[0,1]$. Points of the configurations are labelled by elements of ...
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explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...
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What is the right (a good) definition for a dual monoid?

Suppose we have the free abelian monoid $S = \{a^m : m \in \mathbb{N}_0\}$ on the set of one element $X = \{a\}$. The binary operation on the monoid is denoted by $\cdot$. If $(T,\ast)$ is another ...
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what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
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what is the classifying space of a monoid

In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid $M$ has a classifying space $BM$. I do not understand this ...
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How to read “realize the mapping $x \cdot -: T \rightarrow T$”

This question is about Category theory for the sciences (by David Spivak). In Exercise 3.1.2.4-a the set $T = \{x \in \mathbb{R} \; | \; 0 \leq x < 12\}$ needs to be defined using a coequalizer. I ...
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MTL algebra 'prelinearity' condition etymology

According to wikipedia the prelinearity condition of a monoidal t-norm logic is expressed as $(x\implies y) \vee (y\implies x) = 1$. As far as I know, the 'pre' prefixed version of a rule or ...
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When is a monoid contained in a group?

As stated in the answer of Is the forgetful functor from groups to monoids right adjoint? , the forgetful functor $U:\mathbf{Grp}\rightarrow\mathbf{Mon}$ has a left adjoint $G$, and Grothendieck's ...
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Cancellative Abelian Monoids II

Following this question I was asking myself if (in a cancellative Abelian Monoid $M$) given three elements $a,\, b,\, c$ for which there exists the least common multiple $m$, it will also exists the ...
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Cancellative Abelian Monoids

Is there an example of cancellative Abelian monoid $M$ in which we may find two elements $x$ and $y$ such that they have a least common multiple but not a greatest common divisor?
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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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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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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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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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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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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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$G$ 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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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.