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Questions tagged [monoid]

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

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How many non-isomorphic algebraic structures (i.e. magmas, monoids, groups etc.) are there with countably infinite order? [closed]

For structures of finite order it seems obvious to me that there are countably infinite in total, by a simple diagonalization argument (starting at all of order 1, then 2 etc.). It is however not ...
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Definition of the comparison functor in category theory

I have a problem in showing that the free-forgetful adjunction $F\dashv U: \mathbf{Mon} \to \mathbf {Set}$ (call $\eta$ the unity and $\varepsilon$ the counity) is monadic, and it seems that the ...
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Prove for a semigroup $S$ that $SeS = SfS$ is equivalent to the existence of $x, y \in S$ such that $xy = e$ and $yx = f$

Let $S$ be a finite semigroup and let $e, f$ be idempotents of $S$. I want to show that $SeS = SfS$ is equivalent to the existence of $x, y \in S$ such that $xy = e$ and $yx = f$. The second direction ...
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Must an infinite sum of zeros be zero?

Let $X$ be an infinite set and $M$ a commutative monoid. Find a function $f \colon \mathcal{P}(X) \to M$ such that $f(\emptyset) = 0$ for each element $x$ of $X$, $f(\{x\}) = 0$, for any two disjoint ...
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Projection from n-fold cartesian product to coordinates indexed by a fiber.

Consider the example 1.3.2 (xi) of Emily Riehl (2016) Category theory in context: I am having trouble trying to understand the part where $M^f$ is described. As far as I have understood, $M^f:M\times\...
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an infinite monoid that is not free monoid and does not contain any free monoid [duplicate]

Let $H$ generated by some generators $H=\langle h_1, \ldots, h_n\rangle$ $(n\gt 1)$. My question is whether there exists any monoid $H$ such that $H$, is infinite and $H$ is not a free monoid and $H$ ...
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Let $M$ be a commutative monoid with the cancelation law. Show that an lcm doesn´t exist under these conditions.

Let $M$ be a commutative monoid with the cancelation law and suppose that $a \nsim b, x \nsim y, ax = by, ay = bx$, and $a$ and $b$ are irreducible. A first question was to show that $\gcd(ax,bx) = \...
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What art exists about classification of monoids? [duplicate]

For groups, there is a solid foundation to classify them https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups What art exists for monoids?
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Showing star-freeness of recursively defined languages

Problem: Define a sequence of languages on $A$, a finite alphabet as $D_0 = 1$ (empty string) and $D_{n+1}= (aD_nb)^*$. Show that each $D_i$ is star-free (for each there is an equivalent star-free ...
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Embedding a semigroup into a monoid

I have just started learning about groups and rings and I'm stuck on one exercise. I don't understand what $S^u$ really is and don't know where to start. So if anybody could help me with it, it would ...
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infinite monoid H that is not a free monoid and contains a free monoid as a submonoid [closed]

Let $H= \langle h_1, \ldots , h_n \rangle$ ($n>1$) be an infinite monoid that is not a free monoid. Does $H$ contain an isomorphic copy of a free monoid as a submonoid? EDIT. It is a natural ...
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Finitely generated submonoids of commutative cancellative gcd monoids

Let $(M,\cdot,1)$ be a commutative cancellative gcd monoid. That is, $\forall a,b\in M\colon a\cdot b =b\cdot a$ $\forall a,b,c\in M\colon (c\cdot a = c\cdot b)\implies (a=b)$ $\forall a,b\in M\colon ...
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Endomorphism monoid of a product

Given a monoid $M$ we define the endomorphism monoid $\mathrm{End}(M)$ of $M$ to be the monoid of homomorphisms $M\to M$ along with composition as the operation. My query is whether or not the ...
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Good book for self-study of Magmas/Semigroups/etc.?

I'm currently an undergrad in my second semester of Abstract Algebra. We've covered groups, rings, fields, all that fun stuff. I'm working with Shahriari's "Algebra in Action" as well as ...
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Product of non-idempotent elements in a commutative monoid

Assuming we have a commutative monoid $(M,\cdot)$ such that the non-trivial elements have no inverse. In addition, M contains no non-trivial idempotents. Considering two non-trivial elements $a_1$ and ...
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Monoidal categories that are not symmetric

All the usual examples of monoidal categories that one comes across ($\mathbf{Set}$ with $\times$ as product, $R-\mathbf{Mod}$ with $\otimes$ etc.) are symmetric. Does anyone know an example, ...
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Clarification question about a monoid being a group

I'm reading Jacobson's Basic Algebra and going over some old exercises. One of them says Let M be a monoid generated by a set S and suppose every element of S is invertible. Show that M is a group. I ...
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Centralizers in a monoid of transformations.

This comes from N. Jacobson's Basic Algebra 1, 1.4 #1: Let $A$ be a monoid, $M(A)$ the monoid of transformations of $A$ into itself, $A_L$ the set of left translations, $A_R$ the right translations. ...
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Identity elements of semirings

When we define a group the identity element can be any suitable one. E.g. $(\mathbb{N}, 0, +, -)$ or $(\mathbb{N}, 1, \cdot , N_i^{-1})$ are two groups with $2$ different identity elements. Now it is ...
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Is all algebraic system is monoid?

Is all Algebraic system is monoid? I cross-checked the properties of both monoid and algebraic systems. Here is what I found: Properties of Algebraic system: 1.closure property 2.Associativity 3....
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What is a monoid in simple terms?

I encountered the term "monoid" but I didn't really understand what is it useful for or what's it about. If I understand correctly a "monoid" is something defined in the context of ...
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A finitely based monoid whose semigroup reduct is not finitely based, and vice versa

Does there exist a monoid $(M,*,1)$ which is finitely based, but whose $\{*\}$ reduct is not finitely based? Also, does there exist a monoid $(M,*,1)$ which is not finitely based, but whose $\{*\}$ ...
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“Near equivalence” of semigroups and monoids

Given any semigroup $S$ we can uniquely extend it to a monoid $M$ by introducing a new identity element (even if $S$ already has an identity we can still add a new identity). Conversely, given a ...
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Which operator $\oplus$ turns a function $f$ into a monoid homomorphism from $(L,*,[])$ to $(\mathbb{R},\oplus,\epsilon)$?

This question is inspired by this question. I do not have a strong background in abstract algebra, so this might be trivial or pointless. Let $L$ be the set of all lists of finite length and denote $*$...
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Biprefix code and word factorization

Let $Y_1$ a biprefix code over a free monoid $A^{*}$. Let $u= x_{i}y_{j}$ and $v= x_{i}’y_{j}’$, with $x_i, x_i’ \in A^*$ and $y_j, y_j’ \in Y_1$. If $u=v$, while does this imply that $y_{i} = y_{i}’$...
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How to understand the Grothendieck group as an adjoint?

My main goal is to understand the Grothendieck group construction and its generalization to non-commutative setting. I understand its explicit construction via an equivalence relation, but I want to ...
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Suspicious diagrams on wiki about group-like structures

It seems to me that the diagrams on wiki about group-like structures are not quite right. For example, the following https://en.wikipedia.org/wiki/Monoid#/media/File:Algebraic_structures_-...
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Alternative proof that probability of empty space is 0

In my book, and everywhere the writter found in internet, to prove that $P(\emptyset)=0$, they do this: $1=P(Ω)=P(Ω∪∅)=P(Ω)+P(∅) \implies P(∅)=0$ because $Ω∩∅=∅.$ When the writter tried to prove this, ...
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Natural partial order of Mitsch on Natural numbers

So according to Mitsch, the natural partial order $\leq$ of any semigroup $S$ is given by $$a \leq b \iff a = xb = by, xa = a, \quad \text{for some } x, y \in S^1$$ But obviously in $(\mathbb N, \cdot)...
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Semilattice of the Left Inverse Hull

This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc. A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, ...
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1answer
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“Extending” monoid action on monomials to the entire polynomial ring

Let $X$ be a set of variables, and consider the polynomial ring $K[X]$ in the variables $X$ over a field $K$. Let $Mon(K[X])$ denote the set of monomials of $K[X]$. Let $M$ be some monoid acting on $...
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How can we extend the notion of unique factorization monoid to arbitrary products?

We have a precise definition of unique factorization monoid that concerns products of finitely many elements. How can we extend it to arbitrary products, as the ones that we find in McKenzie's article ...
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Is $ \mathbb{Z}^+ [ x ] \backslash \{ 0 \} $ a unique factorization monoid with respect to the usual product?

Let $ \mathcal{M} $ be the algebraic structure that consists of $ \mathbb{Z}^+ [ x ] \backslash \{ 0 \} $ equipped with the usual product. Let $ M = \mathbb{Z}^+ [ x ] \backslash \{ 0 \} $. We have ...
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Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$. [duplicate]

Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$. Attempt: Does this approach true? $(\Rightarrow)$ Let $S$ be a group. ...
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Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$. [duplicate]

Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$. For the right direction, since $G$ is a group with identity, then there ...
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Groups and Monoids in a Category $\mathcal{C}$?

By slightly generalizing the definition from Awodey's Category Theory, I obtained: (this is almost exactly what is on Awodey 75-76 with a few slight modifications). And the first diagram is the same ...
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Natural Choice of Topology for Free Monoid on a Space

Suppose $X$ is a metric space, and let $X^*$ denote the free monoid on $X$, that is the monoid consisting of all finite strings of elements of $X$, with string concatenation as the monoid operation (...
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Is the product of two elements in a monoid also an element of that monoid? [closed]

Let $(M, ·)$ be a monoid and $x$ and $y$ any two elements of said monoid. Is the following true? $$∀ x, y ∈ M, \qquad x · y ∈ M$$
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Interchanging products for commutative monoids In Serge Lang’s Algebra

In Lang’s Algebra on pages 5-6 he mentions the property involving commutative monoids: “Let $I, J$ be two sets, and $f:I\times J \rightarrow G$ a mapping into a commutative monoid which takes the ...
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Is $\left(K, - \right)$ a group, semigroup, or monoid?

My professor asked us to determine whether $\left(K, - \right)$ for $K$ is a set of integers is a group, semigroup, or monoid. Using my limited knowledge in the group theory here's 4 condition that I ...
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Is there any reference for the optimal bound $3|M|-1$ in Simon factorization forest theorem?

Let $S$ be a finite semigroup. A factorization tree of a product $w_1\cdots w_n$ in $S$ is a finite ordered tree labeled by elements of $S$, such that each leaf is labeled by the $w_i$ in their order ...
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Converse to a theorem regarding monoids and their set of invertibles

In this paper, there is a theorem regarding monoids and their set of invertible elements. Let $(M,*,1)$ be a monoid, and let $U$ be the set of invertible elements of $M$. The theorem states that if $...
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Can the hypothesis in this theorem on commutative monoids be weakened?

Let $(M,*,1)$ be a commutative monoid. Define the binary relation $R$ on $M$, such that $xRy$ iff $(\exists z)(x*z=y)$. It is easy to show that, since $M$ is a commutative monoid, the relation $R$ is ...
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Can a certain monoid exist?

Is it possible to have an uncountable commutative monoid, where for every $a$ in the monoid, $a+a=a$? I have a set which I am trying to define a group structure on (I am settling for a monoid ...
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Is every monoid isomorphic to a submonoid of a full transformation monoid?

We know that every group is isomorphic to a subgroup of a symmetric group. So, the question arises, is every monoid a submonoid of a full transformation monoid, where a full transformation monoid is ...
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From monoids to groups

I was looking at the case when you go from the monoid of natural numbers to the group of integers by means of a suitable equivalence relation. The key here was to find inverses for each natural and I ...
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1answer
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What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (like a polynomial ring)?

I don't think there is much for me to elaborate beyond the title question: "What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (...
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An abelian group proof with $g*g=e$ for all $g$. [duplicate]

I have to show that the following group $$ (G, * , e) $$ with its operation $*$, which is defined through $ g*g = e$ for every $g \in G $ is an abelian group. In order to do that one have only to ...
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1answer
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Right invertible elements in a monoid.

Prove that if every element in the monoid is right invertible, then every element has exactly one right inverse. That is in the monoid $(M,\circ )$ with the identity element e, $$\forall a\in M \; \...
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Doubt about category theory exercise on Bool monoid

I'm studying Category Theory for Programmers. At the end of chapter 3, the exercise number 3 asks Considering that Bool is a set of two values ...

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