Skip to main content

Questions tagged [monoid]

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

Filter by
Sorted by
Tagged with
4 votes
2 answers
122 views

Semirings which cannot be extended to semifields

Definitions By a commutative $\textit{semiring}$ (with 1 and without 0), I mean a triple $(S,+,\cdot)$ where $(S,\cdot)$ is a commutative monoid, $(S,+)$ is a commutative semigroup, and $\cdot$ ...
Antoine de Saint Germain's user avatar
1 vote
2 answers
39 views

Congruence relation in monoids

Suppose $(M, \ast)$ is a monoid and $\sim$ is a congruence relation on $M$, i.e. $\sim$ is an equivalence relation such that: $x_1 \sim y_1$ and $x_2 \sim y_2$ imply that $x_1 \ast x_2 \sim y_1 \ast ...
Douglas's user avatar
  • 414
2 votes
2 answers
45 views

Does the monoid of non-zero representations with the tensor product admit unique factorization?

Let $(M, \cdot, 1)$ be a monoid. We will now define the notion of unique factorization monoid. A non-invertible element in $M$ is called irreducible if it cannot be written as the product of two other ...
Smiley1000's user avatar
  • 1,649
11 votes
1 answer
238 views

The “Pumping Lemma” For Finite Monoids

Let $M$ be a finite monoid. I’m trying to prove the following: there is a constant $N$ such that if $n \geq N$ and $m_1, \ldots, m_n \in M$, then some subword of $m_1 \cdots m_n$ is an idempotent. (...
neddo's user avatar
  • 205
0 votes
0 answers
34 views

Characterizing cofinite submonoids of $\langle \mathbb{N}, + \rangle$?

A set $S \subseteq \mathbb{N}$ is a submonoid of $\langle \mathbb{N}, + \rangle$ when $0 \in S$ and $S$ is closed under addition (that is, $m+n \in S$ whenever $m$ and $n$ are). For example, $\mathbb{...
templatetypedef's user avatar
2 votes
1 answer
31 views

two-way monoidal orbit

Let $M$ be a monoid acting on a set $X$. We can define an equivalence relation on $X$ by $x \sim y$ iff $y \in M x$ and $x \in M y$. Given an element $x \in X$ we can let $R_x \subseteq X$ be the ...
Ben's user avatar
  • 579
2 votes
0 answers
37 views

How to prove the free group $F_n$ cannot be generated by $n$ elements as a monoid?

I cannot figure out how to prove that the submonoid of $F_n$ generated by elements $\alpha_1,\dots , \alpha_n$ will never be the whole group. It clearly is possible to generate the free group on $a_1, ...
Zoe Allen's user avatar
  • 5,633
1 vote
0 answers
39 views

Monoid structure on field of rational functions / algebra that is equal to field + monoid

Let $\mathbb F$ be a field. Recall that the field of rational functions $\mathbb F(X)$ can be presented as fractions $\frac{P}{Q}$ where $P,Q\in\mathbb F[X]$ are polynomials (we silently assume ...
Jim's user avatar
  • 538
2 votes
0 answers
36 views

Monoids that produce sequences without runs

I'm not sure how to phrase this question properly, but is there a name for a property of some monoids/operators that, when applied to a sequence, produce a sequence without runs? For example, suppose ...
Adam B.'s user avatar
  • 121
2 votes
0 answers
60 views

Morphisms in Monoids

As I am trying to learn Category Theory (CT) which is very relevant to my line of research, I am coming across the idea that, in CT, we don't have an "internal view" of say a group as made ...
user2167741's user avatar
4 votes
0 answers
56 views

On the Rationalization of Finite Groups

Let $\Gamma$ be the countable set of finite groups identifying isomorphic ones (formal details below). Then, the direct product and the subgroup relation gives $(\Gamma,\times,\leq)$ the structure of ...
K. Makabre's user avatar
  • 1,810
1 vote
0 answers
28 views

Monoid structure on polynomial quotient ring / quotient fields

Let $\mathbb F$ be a field and let $P\in\mathbb F[X]$ be an irreducible polynomial. Write $\mathbb F'=\mathbb F[X]/P$, and note that this is also a field. Let $P'\in\mathbb F[X]$ be any polynomial ...
Jim's user avatar
  • 538
2 votes
0 answers
43 views

Relation between finite abelian monoids and finite abelian groups

I'm reading the first part of the book "Representation Theory of Finite Monoids" by Steinberg on finite monoids and in the section 1.1 he wrote "a finite monoid is a collection of ...
newuser's user avatar
  • 302
0 votes
0 answers
38 views

(sub)monoids of the positive integers under multiplication, with density $0$ in the positive integers, are always multiplicative norms of rings?

Consider integer polynomials of type $"x"$ where we take as imput nonnegative integers. With these nonnegative integer imputs we strictly generate a subset of nonnegative integers ; the set $X$. The ...
mick's user avatar
  • 16.4k
0 votes
1 answer
54 views

Axioms of Field

I am currently exploring the fundamental of field theory, especially, in its connection to monoid and group. One way we can describe a field $\mathbb{F}$ is using the following axioms: F1. $(\mathbb{F}...
rp23's user avatar
  • 3
1 vote
1 answer
38 views

The 'union of factors' comultiplication in a monoid ring?

let $\mathbb{Z}[M]$ be 'the' monoid ring of $M$ over $\mathbb{Z}$; that is to say (if my understanding is right) the set of finite linear combinations of elements of $M$, with product given by $\sum_n ...
Steven Stadnicki's user avatar
0 votes
0 answers
36 views

Zeno's Monoid: Has anyone got a reference for this?

Let F be an ordered field. Let k be a positive element of F. Define a binary operator * on F: x*y = x+y - xy/k Then I claim ([0,k],*,0) is a commutative monoid with k as an absorbing element. Moreover ...
Nicholas Bamber's user avatar
1 vote
2 answers
49 views

Existence of monoidable submagma of monoid which is neither a submonoid nor idempotent?

Does there exist a monoid $(M;*,1)$ and a submagma $S$ of $M$, such that $S$ is not a submonoid of $M$, but $S$ is "monoidable", meaning it contains an identity element, and also, $S$ is not ...
user107952's user avatar
  • 21.4k
3 votes
1 answer
83 views

Are modules for the ring of formal power series representations of a certain object?

Most of my exposure to modules has been over the various forms of category algebras, such as group algebras, path algebras, and incidence algebras. As such, I don't have a lot of exposure to modules ...
tox123's user avatar
  • 1,602
0 votes
0 answers
31 views

factorization into coprimes subordinate to two given coprimes in GCD domain

Let $a,b,c$ be elements in a GCD domain (or just a GCD monoid) $R$. Suppose that $\text{gcd}(a,b)=1$ and $c \ne 0$. If $R$ is a UFD, it is possible to write $c=a'b'$ with $\text{gcd}(a',a)=\text{gcd}(...
Junyan Xu's user avatar
  • 722
5 votes
0 answers
55 views

The collection of maps to a (commutative) monoid is a (commutative) monoid, via Eckmann-Hilton

A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of ...
Matthew Niemiro's user avatar
6 votes
1 answer
63 views

Can $({\Bbb N}, \max)$ be topologized to be a compact Hausdorff monoid?

I am aware of this question and this one, the answers to which show that the natural numbers can be equipped with a compact Hausdorff topology. But what happens if one also requires the operation $$ \...
J.-E. Pin's user avatar
  • 40.7k
1 vote
0 answers
72 views

Reference request: monoids on ordinal numbers

It is well-known that $(\text{Ord},+,0)$ and $(\text{Ord},\cdot,1)$ are monoids, but I haven't found references on these structures or other simpler ones (like $(\omega_1,+,0)$). For example, it would ...
Yester's user avatar
  • 414
2 votes
1 answer
52 views

Is the category of monoids $\textsf{Mon}(\mathcal{C})$ in a monoidal category $\mathcal{C}$ itself monoidal?

I have read about the forgetful functor $U$ from $\textsf{Mon}(\mathcal{C})$ to $\mathcal{C}$ (see e.g. this nLab page). I think I have read somewhere that this functor is monoidal, but I cannot find ...
user11718766's user avatar
0 votes
0 answers
37 views

What is the relation between Segal Objects and the Lawvere Theory of Commutative Monoids

I am trying to wrap my head around the different ways to describe commutative monoids in categorical algebra. The Lawvere Theory of commutative monoids can be described as the opposite of the full ...
Jonas Linssen's user avatar
2 votes
1 answer
47 views

Can we categorify a monad $T A$ for a fixed set?

In Category Theory, a monoid such as $(\mathbb N, + ,0)$ can be "categorified" as a category of a single object $\mathbb N$, and morphisms as elements of $\mathbb N$. Now, consider a monad $(...
Davi Barreira's user avatar
0 votes
1 answer
28 views

Monoid morphisms between naturals with multiplication and naturals with addition

We define $\mathbf{N} = \{0,1,2,...\}$ and $ \mathbf{N}^* = \{1,2,...\}$, each a monoid with addition and multiplication respectively. I am looking for monoid morphisms between these two monoids. For ...
Jason's user avatar
  • 692
3 votes
1 answer
91 views

The minimal ideal of a finite semigroup whose idempotents commute

In J. E. Pin: On Reversible Automata, LATIN 92, Springer LNCS 584, 1992 the author states that "it is a well-known fact of semigroup theory that the minimal ideal of a semigroup in which the ...
stefan.hetzl's user avatar
2 votes
1 answer
103 views

Free Commutative Monoid Quotient by Relations?

Say I have a commutative monoid $M$ that is generated by three elements $A,B,C$, where I have that $A+C=2B$. I want to write this a free (does that even mean anything?) monoid $\mathbb N^3$ with ...
Chris's user avatar
  • 3,431
0 votes
0 answers
82 views

Nomenclature for a unital magma together with a monoid

Is there some established name/nomenclature for structures $\mathfrak{A} = (A,\, {\oplus},\, {\odot})$, where $(A,\, {\oplus})$ forms a (commutative) unital magma (in particular not associative!), $(...
blk's user avatar
  • 281
3 votes
1 answer
106 views

Identifying group of units of monoid to a point

Let $S$ be a monoid. Let $\mathfrak{g}(S)$ be the group of units of $S$, and denote by $S/A$ the quotient $S/\sim_A$ where $\sim_A$ is the smallest congruence such that $a\sim_A a'$ for all $a, a'\in ...
Jakobian's user avatar
  • 10.5k
0 votes
0 answers
65 views

Why a monoid has only one object but many arrows different from identity?

In Category Theory, a monoid is defined as "a category with one object [...] thus determined by the set of all its arrows, by the identity arrow and by the rule for the composition of arrows"...
Moonlanders's user avatar
1 vote
1 answer
28 views

Two almost disjoint submonoids whose union is the whole monoid

Does there exist a monoid $(M;*,1)$ which has two submonoids $M'$ and $M''$, such that neither $M'$ nor $M''$ is equal to $M$, and neither $M'$ nor $M''$ is the trivial monoid $\{1\}$, the ...
user107952's user avatar
  • 21.4k
2 votes
1 answer
56 views

Standard terminology to refer to the monoids of small order

There are two monoids of order $2$ up to isomorphism: \begin{array}{|c|c|c|} \hline & e & a \\ \hline e & e & a \\ \hline a & a & e \\ \hline \end{array} and \begin{array}{|c|c|...
Robin's user avatar
  • 3,940
0 votes
1 answer
49 views

Subgroups in semigroups vs monoids

Let $S$ be a semigroup (i.e. $S$ is endowed with an associative operation). With some work, one can prove that the idempotents of $S$ are in one-to-one correspondence with maximal subgroups of $S$: ...
mathfan24's user avatar
  • 612
1 vote
0 answers
32 views

Is there a name for monoids where this set is finite [duplicate]

Let $M$ be a monoid and write its multiplication as $+$. Some monoids have the property that for each $k\in M$ the set $\{(i,j) \in M\times M \mid i + j = k\}$ is finite. Is there a name for this ...
N. Virgo's user avatar
  • 7,379
2 votes
1 answer
55 views

Monoid Element with Unique Left Inverse But No Right Inverse?

Can an element of a noncommutative monoid have exactly one left inverse but no right inverse, or is it like in rings, where this is not possible? If this is possible, what is an example of such a ...
MathNeophyte's user avatar
1 vote
0 answers
15 views

Separating a primitive word of $A^*$ from its proper prefixes by a monoid morphism from $A^*$ to $\mathbb Z$.

This question came up as a side issue during the course of a research project and I am wondering whether the answer is yes or no. A word is primitive if it is not a proper power of a shorter word. A ...
J.-E. Pin's user avatar
  • 40.7k
4 votes
1 answer
219 views

Simple examples of comonoids

In textbooks on Category Theory, monoids pop-up all over the place, we several easy examples such as integers, lists and so on. I was then wondering about comonoids. What are some "simple" ...
Davi Barreira's user avatar
4 votes
1 answer
279 views

Is this Monoid Finitely Generated?

Let $G$ be a finitely generated abelian group, $H\leq G$, and $S = \{g_1, \dots, g_n\}$ be some finite generating set for $G$. Let $G’$ be the set of negative-free linear combinations of elements of $...
Teddy Astor's user avatar
5 votes
1 answer
249 views

When does a monoid admit a ring structure?

In this question, it is asked when an abelian group admits a ring structure; that is, if $(G,+)$ is an abelian group, then under what conditions is there a binary operation $\cdot$ on $G$ such that $(...
Joe's user avatar
  • 20.7k
1 vote
3 answers
72 views

If a submonoid is in the union of two submonoids, is it in one of them?

It is well-known that if a subgroup of a group is contained in the union of two subgroups, then it is contained in one of them. I would like to know if the same fact is true for submonoids. So, if $M$ ...
Nulhomologous's user avatar
0 votes
0 answers
64 views

definition of group completion of semirings

I know the group completion of a monoid. If we have a semiring $R$, which is in particular a monoid. Then the group completion of $R$ is an abelian group. But how can we define the completion of a ...
Ziqiang Cui's user avatar
3 votes
1 answer
120 views

Category of Non-Negative Integers and Addition

Following is my attempt to construct a category and some questions: Consider the set of integers $\Bbb{Z}$ as objects. There is a morphism $A \to B$ if there exists some number $c$ such that $A + c = ...
Paul Johnecheck's user avatar
0 votes
1 answer
49 views

How to prove that (ℤ\{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023}, ×) is a monoid?

How to prove that $(\mathbb{Z}\backslash\{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023\},\times)$ is a monoid? Which numbers need to be verified to satisfy closedness for multiplication? Not ...
Eufisky's user avatar
  • 3,267
3 votes
1 answer
142 views

Monad of possibly infinite lists

It is well-known that if $T\colon\mathrm{Sets}\to\mathrm{Sets}$ is the monad which takes a set $S$ to the set of lists of elements of $S$, i.e., $\bigsqcup_{n\ge0} S^n$, with the monad structure $\mu\...
Kenta S's user avatar
  • 16.8k
0 votes
1 answer
54 views

Is the free monoid on $n$ generators just the semigroup on $n$ generators adjoined with an identity element?

Is the free monoid on $n$ generators isomorphic to the semigroup on $n$ generators, just adjoined with an identity element? That is, to get the free monoid, you just take the free semigroup, and then ...
user107952's user avatar
  • 21.4k
0 votes
1 answer
88 views

Product of non commutative periodic elements which is NOT periodic [closed]

There is a proposition which says: If $a$ and $b$ are two periodic elements of a monoid $(G,\cdot\,,1)$ such that $a \cdot b = b \cdot a$, then $a \cdot b$ is a periodic element of the monoid $(G,\...
John Pi's user avatar
  • 153
2 votes
1 answer
99 views

Faithful representations of the bicyclic semigroup (bicyclic monoid)

The bicyclic semigroup $B$ is the semigroup with two generators $p, \; q$ and the single relation that $pq = 1$. So all other words in $B$ are of the form $q^{n}p^{m}$ for $m, n \in \mathbb{N}$. I ...
Dash Stander's user avatar
2 votes
0 answers
71 views

Solution check and question clarification request about a question on "generalized monoid" in Category theory.

The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes In chapter 7 of Arbib and Manes about Functors. The authors introduce the category ...
Seth's user avatar
  • 3,683

1
2 3 4 5
18