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7
votes
1answer
387 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
4answers
477 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
2answers
297 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
2answers
416 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 ...
21
votes
0answers
657 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
2answers
336 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
1answer
167 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 = ...
10
votes
2answers
485 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
2answers
600 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
0answers
381 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 ...
7
votes
2answers
608 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
2answers
310 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
2answers
301 views

submonoids of Z_n

Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative monoid?
3
votes
0answers
257 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
2answers
645 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
1answer
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
2answers
395 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 ...
15
votes
1answer
556 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 Buntz and Moerdijk have shown that if the topos has ...
4
votes
0answers
235 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
2answers
514 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
0answers
707 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
1answer
402 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
1answer
257 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 $A$ ...
14
votes
0answers
670 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
0answers
258 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
1answer
197 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 ...
5
votes
1answer
363 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
2answers
519 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
1answer
400 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
1answer
222 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
1answer
407 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
1answer
130 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
12answers
2k 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
3answers
512 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
1answer
973 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
2answers
637 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
1answer
686 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
2answers
481 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
2answers
466 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
5answers
636 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. ...
50
votes
11answers
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
2answers
354 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
3answers
513 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
1answer
332 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
2answers
362 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 ...
1
vote
1answer
240 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
0answers
807 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
1answer
189 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
3answers
820 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, ...
5
votes
4answers
685 views

Torsors for monoids

Torsors are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful. In general I'm interesting in the ...