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

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Is there an accepted notation for the monoid of linear polynomials?

Is there an accepted notation for the monoid of linear polynomials (with addition as the operation) with coefficients from some ring R? Like $2p+3$, where $p$ and the identity generate the monoid ...
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Universal enveloping group

The universal enveloping group of a monoid (with identity) is a well-known construction. If $A$ is a totally ordered set without a maximal element and $M(A)$ is the monoid of all increasing functions $...
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Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

[Update: I've now asked the same question on mathoverflow.] For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now suppose ...
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Different right / left identity and two sided identity element

Maybe this will be a trivial question but I can't find a solution for this. I checked already on a lot of questions here and on google but can't find a solution for my specific problem. Given an ...
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Monads in monoids

This question is almost a duplicate of this one, but not quite. There the person asked about examples and intuition, I am asking about terminology and applications, and I am addressing my question ...
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Is there a monoid structure on the set of paths of a graph?

Given a graph G, and the set of paths in G called PathG. Is there a monoid structure on PathG? Will concatenation be the multiplication formula? even if it's not defined for some paths? What about ...
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The Krull-Schmidt-Remak Theorem for Semigroups and Monoids

For finite groups, the Krull-Schmidt-Remak-Theorem holds, i.e. if $$ H_1 \times H_2 \times \ldots \times H_k \cong G_1 \times G_2 \times \ldots \times G_l $$ where the $H_i, G_i$ could not be further ...
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Monoids in which $aN = Na$ and $ab \in N \leftrightarrow ba \in N$ aren't equivalent.

Proposition. Let $G$ denote a group, and $N$ denote a subset of $G$. Then the following conditions are equivalent: $aNa^{-1} = N$ $aN = Na$ $ab \in N \leftrightarrow ba \in N$ Proof. ...
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Visualizing co-Yoneda lemma

I am studying the Yoneda and co-Yoneda lemmas, and in order to understand them well I am trying to develop particular cases. This one is getting me in trouble (it must be easy, but I cannot get it): ...
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164 views

Invariant vectors of $A^n B^m$ with $A,B$ orthogonal matrices

Let $A$ be the following matrix:$$A=\dfrac{1}{2}\ \left( \begin{array}{cccccccccc} -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 &...
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21 views

For formal languages $U,V \subseteq X^{\ast}$, what is $\min(U\cdot V)$

Let $L$ be some language, and consider the operator $$ \min(L) := \{ u \in L \mid \mbox{no proper prefix of $u$ is in $L$} \} $$ where a word $u$ is called a prefix of $w$ if it is an initial segment ...
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14 views

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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24 views

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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1answer
23 views

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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36 views

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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42 views

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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29 views

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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65 views

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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31 views

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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39 views

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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34 views

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. Let'...
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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" as ...
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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 $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ $$h(x)=\inf\left\{\...
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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 way ...
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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 ...