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

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Is the category of monoids cartesian closed? Why?

Is the category of monoids cartesian closed? Why? I read Steve Awodey's "Category Theory", and could not solve the exercise in chapter 6, stated above. Here I speak of the "category of monoids" ...
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Monoids and groups

everybody. I got this exercise from Jacobson. Let $M$ be a monoid generated by a set $S$ and suppose every element of $S$ is invertible. Show that $M$ is a group. Proof: every element of $M$ has ...
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How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
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28 views

Left vs right projective resolutions and homology of monoids

Let me use the ad hoc notation $\mathbb Z^l$ and $\mathbb Z^r$ to distinguish between left and right modules. These are trivial modules. The homology of a (discrete) finite monoid $M$ with ...
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Are “$S$-monoids” known to be good for anything?

I came up with the following... ...Definition. Let $(S,\wedge,1_S)$ denote a fixed but arbitrary monoid. (In the examples I have in mind, $S$ is always commutative and idempotent. But ...
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95 views

Factorization of conjugacy equation's solutions in Monoids

01-28 Update: In the first version I was claiming that the authors were not explicitly or implicitly but I was wrong so I change my question [long explaination at the end of the question] Two ...
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51 views

A rather special monoid

While implementing an embryo of computational algebra on my blog I ran into a rather special monoid and I wonder if it's studied before. After implementing a very simple concept of dynamic sets I ...
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Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
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Representation of regular languages by monoids [closed]

I'm interested in representation of regular languages by monoids, and in particular of how to use this kind of representation to get a recognizer. I have found some references on the web, but does ...
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64 views

initial algebra and free monoid

I am missing something to prove that the initial algebra $A^*=\mu x. Fx$ of the functor $FX=I+A \otimes X $ is the free monoid in a monoidal category. Here's one start Summing up I can build 2 ...
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49 views

Is $(\mathbb{N}, x^y)$ a monoid?

Is $A = (\mathbb{N}, x^y)$ a monoid or a group? I think the identity element in $A$ is $e = 1$. So it should be a monoid but I know it can't be a group as the inverse $a$ of is $0$. Is my ...
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28 views

Proof of whether a specific element is in a commutative, invertible monoid.

M is a commutative monoid with identity element denoted by e. U(M) is the set of all invertible elements of M. Let a∈M and b∈M. Prove that if ab∈U(M), then a∈U(M). I'm not sure if the ...
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40 views

Is a non-commutative, invertible monoid closed under an associative binary operation?

I've been given this question that's been puzzling me for a while: $M$ is a monoid with identity element denoted by $e$. $U(M)$ is the set of all invertible elements of $M$. Suppose that ...
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1answer
25 views

closed monoidal posets

Let $X$ be a set, regarded as discrete category. if $X$ has structure of closed monoidal category $(X,\cdot,e)$, then it is easy to show that $X$ is a group: since all the morphisms are identities, ...
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30 views

If $M$ is a monoid, is there accepted terminology for those elements $x \in M$ satisfying $xM = Mx$?

Suppose $M$ is a monoid and consider an element $x \in M$. Then we call $x$ central iff for all $m \in M$, it holds that $am=ma$. A vast weakening of this condition is to merely require $xM=Mx$. Lets ...
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40 views

Comparing Free Monoids and Kleene Closures (Stars)

These are going to be a straight-to-the-point questions: What is the difference between a free monoid and a Kleene Closure (Star) when generated by the set $A=\{1\}$? Let $A^*$ be the free monoid ...
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35 views

Monoid homomorphisms to the additive naturals?

Part of something I'm doing research in requires monoid homomorphisms $(\omega\times\omega,+,\bar{0})\to(\omega,+,0)$. (I'm just using component-wise addition for the product monoid). Are there any ...
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1answer
79 views

The set of invertible elements of a monoid is closed under multiplication [duplicate]

Let $M$ be a monoid and let $U(M)$ be the set of invertible elements of $M$. How can I prove that $U(M)$ is closed under the binary operation on $M$, i.e., that that $a \in U(M)$ and $b \in U(M)$ ...
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93 views

“Localizing” commutative pointed monoids

A pointed monoid is a commutative monoid $A$ with a distinguished element $0\in A$ such that $0\cdot A=0$. Morphisms should preserve $0$. If $A$ is a commutative ring or pointed monoid, and $f\in ...
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Monoids where $\operatorname{Hom}(M,M) \cong M$

What are some examples of monoids where $\operatorname{End}(M) \cong M$? Is there a nice characterization of such monoids? E.g., they will necessarily have a zero element, since $\forall x (x \mapsto ...
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Can every monad give rise to a monad transformer?

Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as ...
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Ma -> (a -> Mb) -> Mb … vs … Ma -> (a -> b) -> b

I am a newbie to the functional programming world. Is there a reason why in all example I found so far about Monad (or should I say Monoid) is written as: ...
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Infinite free commutative semigroup

Definition from P.A.Grillet, "Abstract Algebra" (quoted with modifications): The free commutative monoid on a finite set $X=\{x_1, x_2,..., x_n \}$ is the semigroup of all monomials $x^{a_1}_1 ...
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53 views

(N,+) and (N,*) aren't isomorphic

Prove that the monoids $(\mathbb N,+)$ and $(\mathbb N,\cdot)$ aren't isomorphic. I tried that by assuming that there is an isomorphism f between (N,+) and (N,*). Then f(x+y)=f(x)*f(y), for every ...
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Convention for generators of monoids - is $ \left\{x^n \right\} _{n\geq 0}$ freely generated by $x$?

I'm trying to understand whether the multiplicative monoid $ \left\{x^n \right\} _{n\geq 0}$ freely generated by $x$ or $1,x$. So for monoids, are "zeroth powers" included in generating sets or not?
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What kind of structure is this: finite sequences with concatenation and multiplication.

Let $(M, \times_M)$ be a monoid, call $M^*$ the finite sequences of elements of $M$. For two sequences $u$, $v$, and an element $m \in M$, define: $u+v$: the sequence consisting of the elements of ...
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How can you use Green's relations to learn about a monoid?

Without having ever formally learned any real monoid theory, I was recently pointed to Green's relations by a friend. It sounds like they're quite useful, and I get the impression they might do what ...
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Characterize kernels of monoid homomorphisms

The kernel of a monoid homomorphism $f : M \to M'$ is the submonoid $\{m \in M : f(m)=1\}$. (This should not be confused with the kernel pair, which is often also named the kernel.) Question. Which ...
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If $G$ is generated by $A$, then $f(G)$ is generated by $f(A)$

Let $G$ be a monoid that is generated by the subset $A$. Let $f\colon G\to H$ be a homomorphism between two monoids. Prove that $f(G)$ (the image of $f$) is created by $f(A)$. I know as a fact ...
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Are there any non-trivial automorphisms on the Natural Numbers under addition?

I'm curious about whether or not there is an automorphism on $(\mathbb{N};+)$ that isn't the identity. I suspect there isn't, but I'm not quite sure how to prove it.
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32 views

Show that order of a Group is 2

For a group $(G,*,I)$ if $a,b \in G$ such that $a * b$ has order 2. I need to prove that $b * a$ has order 2. My Work:- $a * b$ has order 2 means $(a * b)^2 =I$ , so i need to prove is $(b * ...
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What is the correct term for a module but with the scalars being elements of a monoid?

From my understanding, a module is similar to a vector space over a field, except the scalars come from a ring. What about the cases when the scalars come from a monoid? Is there a specific term used ...
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Is there accepted notation to distinguish $A^2 = \{a^2 \mid a \in A\}$ from $A^2 = \{ab \mid a,b \in A\}$?

Let $M$ denote a monoid. Then given a subset $A$ of $M$, we may be interested in one or both of the following subsets of $M$: $$\{a^2 \mid a \in A\}, \qquad \{ab \mid a,b \in A\}$$ Both could ...
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Is the space of monoid homomorphisms on a free monoid free?

I'm not sure about this since if you have two free monoid homomorphisms: $f, g : M \to M$, then $f \circ g = h$, $h$ hiding the information about the two maps, in other words we don't always have to ...
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How do you generalize “loops” from usual topology $I:[0,1] \to X$

$I:[0,1] \to X$ being continuous and $I(0) = I(1)$ is usually how one defines a loop in a topological space $X$, but what if your space $X$ isn't typical. For instance what if $X$ is subspace of a ...
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Topology on a free monoid using regular languages.

A free monoid together with arbitrary unions of regular language subsets forms a topological free monoid. Every free monoid homomorphism is continuous with respect to the topology described in 1. ...
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1answer
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Elements of monoids are equal if relatively prime powers of the elements are equal

Assume that $a$ is left cancelable monoid. That is $ab=ac$ implies $b=c$ Prove that if $a^m=b^m$ and $a^n=b^n$ such that $m$ and $n$ are relatively prime, then $a=b$. Here is what I've done so far ...
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Finite monoid has an invertible element

If $M$ is a finite monoid and $au=bu$ in $M$ implies that $a=b$, show that $u$ is a unit. Hint: If $m=\{a_1,\cdots, a_n\}$ show that $a_1u,\dots,a_nu$ are distinct. A unit is defined as an element ...
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Seeking more information regarding the function $\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right].$

Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows. $$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$ The motivation is that according to ...
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Viewing Semigroups as Categories?

I am wondering how to view semigroups as categories. For example, we can easily view monoids as categories with a single object. Unfortunately, semigroups do not necessarily have identities, so the ...
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Is a submonoid of a commutative, finitely generated monoid, always finitely generated? [closed]

Let $M$ be a commutative, finitely generated monoid and $N$ its submonoid. Is $N$ finitely generated as well?
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A monoid where every element has finitely many divisors

Is there a special name or has there been any study of monoids of this form? This came up in considering the general construction of a multivariate power series algebra over a ring $R$; usually we ...
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Length Sets of Numerical Monoids

Suppose we have a numerical monoid $S=\langle g_1,g_2,...,g_n\rangle$. A factorization for an element $n\in S$ is an $n$-tuple $a=(a_1,a_2,...,a_n), a_i\in \mathbb{N}$ satisfying ...
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Does orbit-stabilizer theorem holds for monoid action?

For a group $G$ acting on some space $X$ we know there is a orbit-stabilizer theorem. My question is does this formula holds for monoid action? I think this formula may not hold, as inverse do not ...
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Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
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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 $M ...
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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, ...