All Questions
Tagged with semigroup-theory or semigroups-and-monoids
606 questions
5
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2
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537
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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 ...
- 11.6k
5
votes
1
answer
334
views
Short proof a monoid is a group iff every splitting is right homogeneous
In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...
- 10.5k
5
votes
3
answers
718
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Subsets of $\mathbb{R}^+$ closed under addition
No one's answered the question cumulant problem so here's a simpler question: Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition? In particular, ...
5
votes
1
answer
208
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Conjugacy classes of monoids II: Abelianising a monoid, wrongly
$\newcommand{\unsim}{\mathord{\sim}}$Let $G$ be a group. What is
$$
G/\left(ab\sim ba\ \middle|\ a,b\in G\right)?
$$
Answer: not $G^{\mathrm{ab}}$, but the set of conjugacy classes of $G$.
When ...
- 11.8k
5
votes
2
answers
137
views
The smallest number of vertices for a graph with the same endomorphism monoid
Let $G$ be a directed graph without loop and suppose that $M$ is its endomorphism monoid.
First how we can create a simple graph $\Gamma$ (using $G$), with the same endomorphism monoid? and then how ...
- 237
5
votes
1
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304
views
Sets of natural numbers such that sums of a bounded number of its elements form a semigroup
This is a naive question and I'm afraid it might be better placed on math.se. I would like to leave it to your judgement.
I would like to know what is known about sets $A$ of natural numbers such ...
- 1,445
5
votes
1
answer
655
views
an example of a semigroup with solvable word problem but unsolvable power problem
We say that a semigroup $S$ has solvable power problem if there is an algorithm that takes as input an element $s \in S$ and decides whether or not there exist $m,n \in \mathbb{N}$ with $m \neq n$ and ...
- 549
5
votes
2
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478
views
Generalization of the concept of a measure
Consider the following generalization of the concept of a measure:
Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice.
Let $M = (Y, \bullet, e)$ be a commutative monoid.
An $(L, M)$-measure is ...
- 2,203
5
votes
1
answer
152
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Cartan matrix of the full transformation monoid ring
Let $T_n$ be the full transformation monoid of an $n$-set and $A_n=KT_n$ its monoid algebra over the complex numbers.
Question 1: Is the Cartan matrix of $A_n$ known? Im especially interested to see ...
- 26.5k
5
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1
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359
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Computations of divisor class monoids
Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".
Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
- 26k
5
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1
answer
597
views
Can every cancellative invertible-free monoid be embedded in a group?
A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product ...
5
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1
answer
170
views
Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?
McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents.
Question: Which homotopy types ...
- 63.9k
5
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2
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236
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Descent of flatness from algebras to monoids
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
- 6,700
5
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1
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383
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Projective resolutions for commutative monoids
What is the right notion of a projective resolution of a commutative monoid?
The category Mon of commutative monoids has plenty of projective (and even free) objects. Indeed, for every set $X$ one ...
- 3,497
5
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1
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284
views
Directed homotopy in the Cayley graph of a monoid
There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some ...
- 190
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1
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378
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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 ...
- 51
5
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1
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142
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On the width of the Catalan monoid and the rank of K-groups of the Furstenberg transformation group
The semigroup algebra of the Catalan monoid is isomorphic to the incidence algebra of $P_n$, where $P_n$ is the poset consisting of subsets of { 1,...,n } where for two subsets $X \leq Y$ if and only ...
- 26.5k
5
votes
1
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303
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A characterisation of faces of rational polyhedral cones
This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
- 6,700
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1
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205
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Topological category of topological monoids / operads
The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...
- 53
5
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1
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342
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Can the trivial module be stably free for a monoid ring?
Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ ...
- 38.6k
5
votes
1
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227
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"Tietze-like transformations" for defining interesting bijections between algebraic structures
Consider the following two definitions of the natural numbers:
The natural numbers are the algebraic structure $\mathbb{N}_1$ generated by one constant, $0$ and one unary function, $S$ (and no ...
- 1,173
5
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2
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252
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Monoid of continuous self-maps of (real) surfaces
Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi_{1}(S),\pi_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of ...
- 6,995
5
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2
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317
views
Proving that a semigroup is regular
In a number of diverse situations of interest to me (mostly associated with something called the abelian sandpile model), one can define a nonabelian semigroup generated by commuting elements $a_1,\...
- 19.7k
5
votes
2
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341
views
Existence of a possible counterexample in automaton semigroups
In an attempt to resolve a question posed by Cain in his paper on Automaton Semigroups (open problem 6.12), I would like to know if there exists a finite semigroup $S$ satisfying the following ...
- 61
5
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2
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332
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Questions on weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called weakly symmetric in case $soc(P)=top(P)$ for every indecomposable projective module $P$ and it is called symmetric in case $D(A) \cong A$ as ...
- 26.5k
5
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2
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402
views
Maximal commuting subsets of $\text{End}(X)$
Let $X$ be a set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. We say that $f, g\in \text{End}(X)$ commute if $g\circ f = f\circ g$, and $S\subseteq \text{End}(X)$ is a commuting ...
- 48.9k
5
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1
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304
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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 ...
- 6,210
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1
answer
498
views
Percolation in Cayley graphs of semigroups.
Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...
- 6,201
5
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1
answer
251
views
Monoid associated to $>2$-player Hackenbush
There is some literature on multiplayer combinatorial game theory, but as far as I can tell none of it follows the line of attack below. I'd love a pointer to a similar approach taken in the ...
- 20.5k
5
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1
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254
views
Examples of Yang-Baxter monoids
Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities:
$(X,\circ,1)$ is a monoid,
$f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$
$x\circ y=f(x,y)\circ ...
5
votes
2
answers
364
views
Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...
- 505
5
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1
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293
views
semigroups acting as continuous functions on regular rooted trees
Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such ...
- 51
5
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0
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191
views
Do most semigroups have a zero?
It is widely believed in finite semigroup theory that asymptotically almost all finite semigroups $S$, up to isomorphism, are 3-nilpotent, i.e., they satisfy $\#\{abc\,:\,a,b,c\in S\} = 1$. My ...
- 151
5
votes
0
answers
187
views
Isbell duality for monoids and groups
Isbell Duality
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Sets}{\mathsf{Sets}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\B}{\...
- 11.8k
5
votes
0
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107
views
Structure of well-ordered commutative monoids
Let $(M,+)$ be a commutative monoid. Let $<$ be a well-ordering on $M$, where
$\forall a\in M,\ 0\leq a$
$\forall a,b,c\in M,\ a<b\Rightarrow a+c<b+c$
The first condition means $M$ will be ...
- 18.7k
5
votes
0
answers
160
views
$S$ and $T$ globally isomorphic semigroups, with $S$ (commutative and) cancellative, iff $S$ is isomorphic to $T$?
Denote by $\mathcal P(S)$ the semigroup obtained by equipping the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication ...
- 10.5k
5
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0
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138
views
Can we define partial group actions on (finite) sets via generators and relators?
Let $G = \langle Y | R \rangle$ be a finitely presented group. A partial group action on a set $X$ is a premorphism into the inverse semigroup
$$
\mathcal I (X) = \{ f: A \to B : A, B \subseteq X, f\...
- 639
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0
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225
views
The forgetful functor from Groups to Semigroups
While teaching this term I found myself reminded of the fact that the "usual" definition of a group homomorphism is really the definition of a semigroup homomorphism, applied to semigroups ...
- 25.8k
5
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0
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107
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Heuristics for the word problem for monoids
The question is about a purely practical problem:
Given is a list of identities, as in http://www.findstat.org/MapsDatabase/Mp00069:
...
- 5,822
5
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0
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64
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Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
- 289
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0
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395
views
Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
5
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0
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99
views
Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
5
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0
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637
views
Unique product groups (and semigroups)
A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
- 18.7k
5
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0
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137
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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 xhytxy=...
- 563
5
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0
answers
295
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 ...
- 615
5
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0
answers
2k
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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 generality,...
- 1,811
5
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0
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350
views
Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,882
4
votes
4
answers
1k
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} ...
- 8,512
4
votes
1
answer
417
views
Is a solvable group satisfying a semigroup law?
Let $S$ be the free semigroup on the set $\{x_1,\ldots ,x_n\}$, where $n$ is a positive integer. Suppose that $\mu=\mu (x_1,\ldots ,x_n)$ and $\nu = \nu (x_1,\ldots ,x_n)$ are two elements in $S$. We ...
- 883
4
votes
2
answers
544
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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 $s_1,...,s_n$ ...
- 41