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

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Monoid operation order sensitive?

It is a basic question, none the less I cannot find an answer: A monoid is associative (with an identity) (m1∙m2)∙m3=m1∙(m2∙m3). e∙m=m∙e=m If you consider a monoid over natural numbers (N,+,0) for ...
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Bi-character for finite, commutative monoids?

If I have a finite commutative monoid $M$ (which is not a group), is it possible to get a bi-character on this? By bi-character, I mean a map $\beta:M\times M\rightarrow \mathbb{C}^*$ such that, ...
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Proof of a Property of a Monoid: Show that the following are equivalent

I am stuck on the following question: Show that the following are equivalent for a monoid M: If $ab$ is a unit, then both $a$ and $b$ are units If $ab = 1$, then $ba = 1$ I am able to show that ...
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Does every surjective morphism from an uncountable into a countable monoid admit a homomorphic right inverse function

Let $M$ be a an uncountable monoid (like $\mathbb R$ with addition or multiplication) and $N$ be a countable monoid (like $\mathbb N_0$, or $\mathbb Z$ with addition or multiplication). Further ...
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Are the free monoids always infinite?

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from ...
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If monoid satisifes universal mapping property over $X$, then $X$ generates the monoid

A monoid $M$ satisfies the universal mapping property (UMP) over $X$, if $X \subseteq M$ and for every map $\varphi : X \to N$, where $N$ is another monoid, there exists a unique homomorphism $\varphi ...
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Specifying a homomorphism by given the images of the generators and extending “lineary”

Let $M$ be a finitely generated monoid with generators $g_1, \ldots, g_k$. Now is every homomorphism $\varphi : M \to N$ uniquely specified by listing the images of its generators? Of course the image ...
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A monoid is universal (or free) over its generators iff no nontrivial relations hold among its generators

Let $X$ be any set. A monoid $M$ is called universal over $X$ iff $X \subseteq M$ and for every other monoid $N$ and function $\varphi : X \to N$ there exists a unique extension $\varphi : M \to N$ of ...
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Proving that Z with the binary operation is a monoid?

Let $*$ denote the binary operation defined on the set $\Bbb Z$ of integers, where $$x * y = 3xy - 5x - 5y + 10$$ for all integers $x$ and $y$. Prove that $\Bbb Z$, with the binary ...
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Game: Group and Multi-Dimensional Chessboard

Let $G$ be a group and $S\subseteq G$. Consider a $d$-dimensional chessboard of size $n_1\times n_2\times \ldots \times n_d$, where $n_1,n_2,\ldots,n_d\in\mathbb{N}$. Each unit hypercube of the ...
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Multiplicative identity in a monoid ring.

Let $R$ be a ring and $S$ a subset of $R$. I want to prove that $1:S\rightarrow R: s \mapsto 1_R$ is the multiplicative identity in the ring $(R^{(s)},*,+,1,0)$ (with $R^{(S)}$ the subset of $R^S$ ...
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Which polynomials make $\mathbb{R}$ into a monoid?

Question. Can we describe the set $$\{P : \mathbb{R}^2 \rightarrow \mathbb{R}, e:\mathbb{R}\mid P\; \mbox{is a bivariate polynomial}, P \mbox{ is associative}, P(e,x) = P(x,e) = x\}$$ explicitly? ...
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Can every monoid action be turned into a group action?

Let $\mathbf{Mon}$ be the category of monoids. Let $\mathbf{Grp}$ be the category of groups. There is the inclusion functor $i : \mathbf{Grp} \to \mathbf{Mon}$. It has both a left and a right adjoint; ...
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Bound on minimal word length by generators of monoid

Say I have a monoid of size $n$ (a group except there aren't always inverse elements), I have some subset and I'm looking at the submonoid (is that a word?) formed by the subset, is there a bound on ...
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The monoid of integers is not free

I am reading the introductory lessons on Category Theory on wikiversity, and they discuss free monoids here: https://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Monoids At the bottom ...
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Grothendieck-Group, formal difference

My question concerns the Grothendieck completion of a abelian Monoid A and the formal difference. I quote from a book: Instead of writing elements $G(A)$ (thats the Grothendieck-Group), it is ...
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Proving a monoid is associative

Let M be a commutative monoid and set $M^+=$ {$a \in M : a^k$ is idempotent for some $k\ge 1$}. Prove that $M^+$ is a monoid with the binary operation induced from M. I have proven $M^+$ is closed ...
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Monoids and Idempotents

If an element of a commutative Monoid M is said to be idempotent for some $k \ge 1$ then I am trying to show that $M^+$ the set of elements that are idempotent is closed. What I understand is that if ...
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How to check whether arbitrary finite [syntactic] monoid is aperiodic or not?

Does there exist an algorithm to decide whether a (finite in my case) syntactic monoid is aperiodic or not? By definition, a monoid is aperiodic if for each $x$ from monoid there exists an $n$ with ...
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What congruences of abelian monoids can be extended to (ideal) congruences of polynomials?

Let us consider the ring $\langle K[x_1,...,x_n],+,\cdot \rangle$ where $K$ is either a field or the ring of integers. It is well known that congruences of this polynomial ring are characterized by ...
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Functor from group viewed as a category to another category.

I was reviewing some topics in Category Theory when I came across "monoid categories". I mean those with a single object $\{*\}$ and with the composition rule given by the product in some monoid. ...
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Example of a monoid having at least 2 members for which $xy = 1$ but $yx \not= 1$

Can a monoid have at least 2 members for which $xy = 1$ but $yx \neq 1$ ? I tried matrix multiplication but $ AX = I$ then $XA = I $ too.
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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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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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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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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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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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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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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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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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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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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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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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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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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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“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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(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 ...