The monoids tag has no wiki summary.

**3**

votes

**1**answer

98 views

### group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...

**1**

vote

**2**answers

153 views

### Are monoids with zero and partial homomorphisms related?

Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U ...

**2**

votes

**1**answer

150 views

### pseudovarieties and profinite group : do * and g() commute?

Let $V$ and $W$ be pseudovarieties of finite monoids. We denote with $gV$ the pseudovariety of categories generated by $V$, and by $V*W$ the semidirect product of pseudovarieties $V$ and $W$.
Does ...

**-2**

votes

**1**answer

110 views

### configuration space and iterated loop space

Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...

**1**

vote

**2**answers

159 views

### What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?

Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...

**2**

votes

**1**answer

112 views

### Finitely generated ordered monoids and noetherian subsets

(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...

**2**

votes

**0**answers

176 views

### Profinite Topology

Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism ...

**1**

vote

**0**answers

57 views

### Name for condition on map of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that
$k(\epsilon)=\epsilon$
for all $a,b\in M$, there exists $v\in N$ such ...

**2**

votes

**0**answers

130 views

### Pro-p topology on free group

Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...

**50**

votes

**11**answers

5k 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 ...

**1**

vote

**1**answer

89 views

### Lax monoids where only the unit triangle is lax

I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...

**3**

votes

**1**answer

61 views

### Cosets of the fixer of an action of a monoid on a finite set

Let $M$ be a monoid that acts transitively from the right on a finite set $X$.
Assume furthermore that the action of $M$ on $X$ induces for every $m \in M$ a bijection on $X \to X, x \mapsto x.m$.
Let
...

**3**

votes

**1**answer

255 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 $A$ ...

**3**

votes

**0**answers

87 views

### Pseudovarieties of monoids

All (pseudo)varieties considered here are (pseudo)varieties of monoids.
It is known that any (finite or infinite) monoid that satisfies the identities
\begin{equation}
xhxyty = xhyxty, \quad ...

**2**

votes

**0**answers

112 views

### Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...

**5**

votes

**3**answers

224 views

### Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...

**2**

votes

**0**answers

169 views

### Standard name for a Monoid/Semigroup with $a+b \leq a, b$?

I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice?
For instance, for reals $a,b > 0$, define $$a \oplus b = ...

**2**

votes

**0**answers

72 views

### A construction on commutative monoids similar to the semidirect product

Let $M_1$ and $M_2$ be commutative monoids, $M_1$ written additively with identity $0$ and $M_2$ multiplicatively with identity $1$. Furthermore, let $M_2$ act on the left on $M_1$ via monoid ...

**1**

vote

**0**answers

53 views

### on reductive monoids which are gorenstein

Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore ...

**10**

votes

**4**answers

846 views

### Why is a monoid with closed symmetric monoidal module category commutative?

Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural ...

**7**

votes

**1**answer

235 views

### Über theorem on unavoidable patterns?

Let $A$ be an alphabet of $k$ symbols,
and $p$ a pattern.
An example of a pattern is $p=XX$, where $X$ is any finite
string of symbols from $A^+$.
Avoiding $p$ is avoiding any subword repeated twice ...

**1**

vote

**2**answers

139 views

### Comonads from monoids

The following construction is probably known. I think it should work in any closed symmetric monoidal category, but I will play it safe and formulate the question in the concrete, cartesian closed ...

**2**

votes

**1**answer

116 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**4**

votes

**1**answer

178 views

### Is the monoid of taking iterated images and inverse images freely generated by the image and inverse image operation?

Let $\mathcal{F}$ denote the class of all functions. Let $U,L:\mathcal{F}\rightarrow\mathcal{F}$ denote the mappings where if $f:X\rightarrow Y$, then $U(f):P(X)\rightarrow P(Y),L(f):P(Y)\rightarrow ...

**3**

votes

**0**answers

134 views

### Orbit-Stabilizer theorem for continuous groups

The orbit-stabilizer relationship (also known as the orbit-stabilizer theorem) is very clear for finite groups. Is there an equivalent relation for continuous groups?
Also, is there a similar notion ...

**0**

votes

**0**answers

74 views

### Monoid action on an uncountably infinite set

The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition ...

**1**

vote

**2**answers

178 views

### “Exactness” of groupify functor

For each commutative monoid $M$, there exists a "groupification" $\widehat{M}$, i.e. an abelian group that satisfies an obvious universal property.
I tried to prove the following: If in the diagram ...

**1**

vote

**0**answers

89 views

### First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...

**4**

votes

**2**answers

174 views

### Embedding a linearly ordered free monoid into a linearly ordered group

A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...

**2**

votes

**2**answers

190 views

### If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...

**1**

vote

**1**answer

364 views

### Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...

**2**

votes

**1**answer

115 views

### Rep of Non-Commutative Monoids

Let M be a non-commutative monoid. It is possible that all representation of M are one dimensional ??
(for groups the answer is negative. Take a non zero x=[a,b]. Take a representation where x does ...

**4**

votes

**0**answers

144 views

### Correspondence between numerical semigroups and polynomials?

A numerical semigroup $A$ is defined as a subsemigroup of the semigroup $(\mathbb{N},+)$ of the positive integers such that the set $\mathbb{N}\setminus A$ is finite. Equivalently (for a subsemigroup) ...

**3**

votes

**0**answers

149 views

### What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...

**5**

votes

**1**answer

331 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

319 views

### Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...

**1**

vote

**1**answer

194 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

**0**answers

120 views

### Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...

**5**

votes

**2**answers

515 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 ...

**14**

votes

**12**answers

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 ...

**2**

votes

**1**answer

96 views

### Maximal sub-inverse semigroups of $M_n(\mathbb{C})$ and $M_n(F_p)$

An inverse semigroup $S$ is a semigroup in which every element $x \in S$ has a unique inverse $y \in S$ such that $x = xyx$ and $y = yxy$. Are there some references characterizing the maximal ...

**4**

votes

**1**answer

178 views

### What is the formula for the commutative multiplication on CP(infinity)?

There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian ...

**4**

votes

**2**answers

277 views

### Concatenation of strings [closed]

We have two strings (i. e., finite tuples) $A$ and $B$.
We have to find if for some positive integers $n$ and $m$, the string $A$ concatenated $n$ times equals the string $B$ concatenated $m$ times or ...

**2**

votes

**1**answer

219 views

### Monoids and groups of fractions

Let $G$ be a group containing a monoid $M$ that spans $G$ as a group. Is it possible to have a proper quotient $\varphi \colon G \to Q$ of $G$ such that the restriction of $\varphi$ to $M$ is ...

**4**

votes

**1**answer

141 views

### Progress on group languages characterizations

Def. A group language is a recognizable language whose syntactic monoid is a group.
q1. Is it known a "nice" combinatorial characterization of group languages ?
q1.1. If no, is it well understood ...

**1**

vote

**2**answers

232 views

### Are orbits of an affine algebraic monoid affine?

Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...

**3**

votes

**0**answers

142 views

### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...

**20**

votes

**3**answers

799 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 ...

**3**

votes

**2**answers

355 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 ...

**5**

votes

**2**answers

163 views

### Functionals on oriented matroids

Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors.
Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...