The monoids tag has no usage guidance.

**21**

votes

**3**answers

942 views

### What are the main structure theorems on finitely generated commutative monoids?

I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...

**7**

votes

**1**answer

411 views

### Question about topological monoid maps

Let Mon be the category of topological monoids. I am happy to work with the model structure mentioned here:
Model Structure/Homotopy Pushouts in topological monoids?.
I'm looking for a reference ...

**4**

votes

**4**answers

499 views

### Why do we choose the standard total order on the integers?

I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} ...

**2**

votes

**2**answers

301 views

### How many idempotent relations are there on an $n$-element set?

As far as I know, it is an open problem to give a formula counting transitive relations on an $n$-element set. Is it easier to count the idempotent relations, that is relations that are both ...

**2**

votes

**2**answers

441 views

### Semiring naturally associated to any monoid?

For any monoid $M$, we can naturally construct a semiring $S$ as follows:
Let the additive monoid of $S$ be the free commutative monoid on $M$
Let the multiplicative monoid of $S$ be $M$
Then, if ...

**22**

votes

**0**answers

737 views

### Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?

For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...

**7**

votes

**2**answers

339 views

### Does every commutative monoid admit a translation-invariant measure?

Let $T$ be a commutative monoid, written additively. The set $T$ is equipped with a canonical pre-order, defined by $s \le t$ when there exists $s' \in T$ so that $s + s' = t$. Consequently, $T$ may ...

**0**

votes

**1**answer

173 views

### Kernel elements for the Grothendieck group map of a commutative monoid

This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = ...

**12**

votes

**2**answers

523 views

### The symmetric monoidal category of finite sets

It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...

**6**

votes

**2**answers

686 views

### What is the free monoidal category generated by a monoid?

In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category ...

**6**

votes

**0**answers

388 views

### Duality between conjugacy classes and irreducible characters for finite monoids?

Qiaochu's answer to this question suggests that the proper way to view the bijection between conjugacy classes and irreducible complex representations of a finite group is via a duality. My question ...

**8**

votes

**2**answers

635 views

### Connective spectra versus simplicial abelian groups - very basic question

Hello,
I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).
I guess that connective spectra have a model ...

**7**

votes

**2**answers

320 views

### Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF?

The title question is too vague so let me be specific.
Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ...

**1**

vote

**2**answers

303 views

### submonoids of Z_n

Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative
monoid?

**3**

votes

**0**answers

264 views

### Is there a homological way to compute quiver presentations?

I have recently been studying with colleagues the representation theory of certain finite monoids that come up in probability theory and combinatorics, see Ken Brown's beautiful survey here.
These ...

**20**

votes

**2**answers

653 views

### Mapping from a finite index subgroup onto the whole group

Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess ...

**5**

votes

**1**answer

276 views

### flat maps of monoids which are not localizations

It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.
Rather, I am looking for extensions of rings which share certain properties of localizations, like ...

**1**

vote

**2**answers

398 views

### Terminology for certain monoids which are to monoids like fields are to rings

Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work ...

**19**

votes

**2**answers

652 views

### Toposes (topoi) as classifying toposes of groupoids

A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...

**4**

votes

**0**answers

243 views

### What is the pro-algebraic completion of the free semigroup on one generator?

This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...

**10**

votes

**2**answers

593 views

### Exact sequence of monoids

What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
...

**3**

votes

**0**answers

776 views

### Is the radical of a homogeneous ideal homogeneous?

Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this ...

**10**

votes

**1**answer

423 views

### Is the following construction of the 0-Hecke monoid (well) known?

Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...

**3**

votes

**1**answer

261 views

### submonoid of a matrix monoid with a common eigenvector

Hello,
I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ ...

**15**

votes

**0**answers

705 views

### Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...

**4**

votes

**0**answers

265 views

### When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE ...

**1**

vote

**1**answer

201 views

### The intersection of Block Groups and R-trivial (finite) monoids

Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the ...

**6**

votes

**1**answer

373 views

### Growth zeta-functions of regular languages

Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are ...

**5**

votes

**2**answers

527 views

### Do all finitely generated nilpotent semigroups have polynomial growth?

The notion of nilpotency passes nicely from groups to semigroups. Define $q_1(x,y)=xy$ and $$q_{i+1}(x,y,z_1,\cdots,z_i)=q_i(x,y,z_1,\cdots,z_{i-1})z_iq_i(y,x,z_1,\cdots,z_{i-1})$$ inductively for all ...

**7**

votes

**1**answer

402 views

### Powers of maps on finite sets

Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in ...

**3**

votes

**1**answer

228 views

### Can cones (toric monoids) be built as colimits of their faces?

Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...

**7**

votes

**1**answer

414 views

### Cones, monoids, and the space of (very) ample divisors

An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...

**0**

votes

**1**answer

134 views

### Small set of acts over a countable monoid?

Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?

**14**

votes

**12**answers

3k views

### Why semigroups could be important?

There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...

**9**

votes

**3**answers

536 views

### The concept “conjugate class” in monoids.

Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.

**8**

votes

**1**answer

979 views

### Give an example of monoid with property $m^2 = m^3$

Give an example of finitely generated, infinite monoid $M$ with property that for all $m \in M$ we've got $m^2 = m^3$.
This question comes from the problem I was given during algebraic languages ...

**10**

votes

**2**answers

646 views

### Adding a formal inverse of an element to a free monoid

Let $FM_2=\langle a,b\rangle$ be the free monoid of rank 2. If we add a formal inverse to the word $aba$, we get the free group $F_2$ (because both $a$ and $b$ will have inverses).
Question: For ...

**7**

votes

**1**answer

718 views

### Lattice-ordered commutative monoids

By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...

**5**

votes

**2**answers

485 views

### If $k[S]$ is noetherian, is S finitely generated?

Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is ...

**4**

votes

**2**answers

475 views

### Invertible elements in monoid rings of unital monoids without non-trivial invertible elements

This question is somewhat related to Tilmans notorious problem in this post. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 ...

**11**

votes

**5**answers

651 views

### Monoids with infinite products

Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. ...

**55**

votes

**11**answers

6k views

### Why aren't representations of monoids studied so much?

It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...

**4**

votes

**2**answers

358 views

### Membership problem in monoids

What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ ...

**1**

vote

**3**answers

525 views

### Finitely generated monoids are finitely presented?

I saw in the answer of this post that any finitely generated monoids are finitely presented in the sense that there is a coequalizer diagram $P_1\rightrightarrows P_0\rightarrow M$ with $P_1$ and ...

**5**

votes

**1**answer

335 views

### Representations of products of groups (and monoids)

I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).
Suppose ...

**3**

votes

**2**answers

395 views

### What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes?

For a monoid $M$ and a subset $S$ of $M$, define the syntactic congruence $\equiv_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. :
$$u \equiv_S v \Leftrightarrow (\forall x, y)[xuy ...

**2**

votes

**1**answer

254 views

### Vocabulary on monoid periodicity

I'm learning about periodic languages, and I'm confused over the vocabulary used to describe the periodicity of (syntactic) monoids.
If I understand correctly, a monoid M is periodic if :
$$(\forall ...

**5**

votes

**0**answers

808 views

### Associative binary operations on natural numbers

Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too ...

**1**

vote

**1**answer

193 views

### $\omega$-monoids

Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of ...

**8**

votes

**3**answers

835 views

### Structure Theorem for finitely generated commutative cancellative monoids?

Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...