All Questions
Tagged with or semigroups-and-monoids semigroups-and-monoids
606 questions
8
votes
1
answer
448
views
Can a Shelah semigroup be commutative?
A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$...
- 41.9k
3
votes
1
answer
128
views
Size of a minimum generating set for full transformation monoids
Given any finite set $X$ the set $\mathcal{T}(X)=X^X$ of all functions from $X$ to $X$ clearly forms a monoid under composition. Now if we call any family of functions $\mathcal{F}\subseteq \mathcal{T}...
- 2,506
6
votes
1
answer
444
views
Homotopy type of a specific discrete monoid
Consider the discrete monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$, since $z$ is ...
- 3,901
9
votes
1
answer
211
views
Reference for Kakutani result on power sum bases of symmetric functions
Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
- 528
5
votes
2
answers
332
views
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
3
votes
1
answer
122
views
Partition theorems for located words
In this paper Bergelson, Blass, and Hindman prove the following
Theorem 1.2 Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. ...
0
votes
1
answer
655
views
How to show two semigroups are isomorphic?
I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the ...
- 233
13
votes
2
answers
316
views
Semigroup of differentiable functions on real line
Let $D(\mathbb R) $ be the set of all differentiable functions $f: \mathbb R \to \mathbb R$. Then obviously $D(\mathbb R)$ forms a semigroup under usual function composition. Can we characterize (up ...
- 235
1
vote
0
answers
116
views
A generalized Cauchy type functional equation
Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...
- 1,209
0
votes
0
answers
213
views
make me idempotent
$T_n$ be the full transformation semigroup on $X_n= \{1, 2, \cdots , n\}$.
$D_r =\{\alpha \in T_n: |im(\alpha)|=r\}$.
$E(D_r)$ is the set of all idempotents of semigroup $T_n$.
$support(\alpha)=\{...
- 109
3
votes
1
answer
189
views
Faces of polyhedral cones and open immersions of affine toric schemes
Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.
Let $\sigma\subseteq V$...
- 6,700
6
votes
0
answers
340
views
Would you like a subject class for semigroup theory on the arXiv?
After contacting the arxiv recently about possibly adding semigroup theory as a subject class, they suggested I canvas the research community to establish whether such a subject class would be used ...
- 77
6
votes
0
answers
117
views
Closedness of the partial order in complete Hausdorff semitopological semilattices
First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
- 41.9k
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
user111492
1
vote
1
answer
106
views
Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
10
votes
2
answers
716
views
On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
- 613
10
votes
1
answer
579
views
Group completion of topological monoids
Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\...
- 1,252
1
vote
1
answer
110
views
Indecomposable monoids
Let $M$ be a commutative reduced and cancellative monoid and $K(M)$ its group of quotients.
We say that $M$ is indecomposable if for every divisor-closed submonoids $M_1$ and $M_2$, $M=M_1\oplus M_2$...
- 933
7
votes
1
answer
235
views
name for monoids inducing bimonoids in Rel?
Let Rel be the category of sets and relations, which is a (compact closed) symmetric monoidal category under the cartesian product of sets. We write $A \nrightarrow B$ to indicate a relation from $A$ ...
- 3,393
3
votes
0
answers
80
views
On the compactification of partial semigroups
We begin by introducing some relevant definitions.
Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...
- 745
-4
votes
1
answer
224
views
Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? [closed]
Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there ...
- 48.9k
1
vote
0
answers
82
views
What is known about the cohomology of the matrix monoid?
When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object).
Let $M_n(R)$ be the monoid of matrices with matrix ...
- 1,726
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
- 38.6k
0
votes
0
answers
136
views
Amalgamated free-product of semigroups (definition)
I am self-studying some concepts including the title one. I reached the definition of an amalgamated free-product ${S_1}{*_U}S_2$ where $[S_1, S_2; U, w_1,w_2]$ is an amalgam of semigroups. Let $S_1=\...
- 233
3
votes
1
answer
129
views
an inverse semigroup (and perhaps a $C^*\!$-algebra) associated with a directed graph
The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^*\!$-algebra whose generators are partial isometries ...
- 379
6
votes
2
answers
183
views
a category associated with an inverse semigroup
Let $S$ be an inverse semigroup. Define a category $C(S)$ as follows:
the objects of $C(S)$ are the elements of $S$
for any $a,b,e\in S$ let $e\colon a\to b$ be a morphism of $C(S)$ iff $aa^{-1}eb^{-...
- 379
1
vote
1
answer
158
views
Semigroups admitting commutative group actions
Let $(S,*)$ be a semigroup admitting a distinguished element $0$ such that $z*s = s*z = z$, for all $s \in S$. Moreover, let $(\mathbb{G},\cdot)$ be a commutative group. Consider an action
$$
\mathbb{...
- 219
1
vote
0
answers
75
views
partially commutative like monoids [duplicate]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\}...
- 1,269
1
vote
0
answers
256
views
partially commutative monoid [closed]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
- 1,269
6
votes
0
answers
81
views
Representing meet-semilattices with vector spaces of specified dimensions
Take $K$ to be a field and take $L$ to be a finite meet-semilattice. I'm interested in the set of functions $n: L \rightarrow \mathbb{Z}^{\ge 0}$ such that there is some function $V$ from $L$ to ...
1
vote
1
answer
134
views
amalgamated sum of monoids
Consider the amalgamated sum $Q_1 \rightarrow^{v_1} Q_1 \oplus_P Q_2 \leftarrow^{v_2} Q_2$ of $Q_1 \leftarrow^{u_1} P \rightarrow^{u_2} Q_2$ with $Q_1,Q_2,P$ being monoids.
Why does $v:= v_i \circ u_i$...
- 65
3
votes
1
answer
167
views
When are all the convolution roots of an infinitely divisible probability measure infinitely divisible?
Let $G$ be a topological group.
Let us say that a probability measure $\mu$ on $G$ is strongly infinitely divisible (SID) if $\mu$ is infinitely divisible and any probability measure $\nu$ on $G$ ...
- 128k
2
votes
0
answers
80
views
Are the roots of an infinitely divisible probability infinitely divisible themselves?
Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too?
A sufficient criterion would be to ...
- 5,407
1
vote
0
answers
255
views
Presentation of amalgamated sum as a quotient of the direct sum
I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...
- 65
0
votes
0
answers
53
views
a generalization of group (monoid with order-by-order invertible elements)
Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
- 2,232
10
votes
0
answers
367
views
A formula for Frobenius number of certain numerical semigroups
The old formula for the Frobenius number of a numerical semigroup generated by two elements can be stated as follows: assume $\gcd\{a+1,b+1\}=1$, then the Frobenius number of $S= \left<a+1,b+1\...
- 30.5k
3
votes
1
answer
233
views
Is there any characterization and/or classification of subsemigroups of finite monogenic semigroups?
A semigroup $S$ is called monogenic if $S$ is generated by some element $a$ (which is unique if $S$ is not a group) in the sense that $S=\{a^n:n\in\mathbb N\}$.
Observe that each mongenic group is ...
- 41.9k
4
votes
1
answer
189
views
Some questions about homogroups
Every semigroup containing an ideal subgroup is called a homogroup. Let $(S,\cdot)$ be homomgroup, hence it contains an ideal $I$ that is also a subgroup. It is easy to see that $I$ is the least ideal,...
- 995
1
vote
0
answers
40
views
A exemple of a strongly-continuous contraction semigroup : how to prove the contraction?
I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is ...
- 111
3
votes
2
answers
249
views
Question about actions of full transformation monoids
[Reposted from math.stackexchange]
Consider a monoid $M$ acting on a set $X$, where $M$ is the full transformation monoid on some set $A$ (i.e., the set of all functions from $A$ to itself, with ...
- 167
8
votes
3
answers
609
views
Nielsen-Schreier theorem for monoids
Let $S$ be a finitely generated free abelian semigroup (or monoid), and let $T \subset S$ be a sub-semigroup (sub-monoid). Does the Nielsen-Schreier theorem hold in this case, that is, will $S$ still ...
- 459
2
votes
0
answers
81
views
A semigroup property related to von Neumann regularity
A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...
- 18.7k
9
votes
2
answers
667
views
Semi group of polynomials which all roots lie on the unit circle
Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $.
The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials.
With the standard multiplication, $X$...
- 356
11
votes
0
answers
214
views
Is it decidable if a tree-presented semigroup contains an idempotent?
A semigroup presentation $\langle A | R\rangle$ is called tree-like if every relation has the form $ab=c$, $a,b,c$ are in $A$ and if two relations $ab=c, a'b'=c'$ belong to $R$, then $c=c'$ if and ...
user6976
2
votes
1
answer
302
views
Name for this algebraic structure?
I've found myself looking at a structure $\mathbb{M}$ whose important properties are:
$\mathbb{M}$ is a discretely ordered additive monoid.
$\mathbb{M}$ has a least element, and this least element is ...
- 10.1k
3
votes
0
answers
92
views
Is the elasticity of a submonoid of the free abelian monoid over a finite set either rational or infinite?
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$, and $H$ a submonoid of $\mathscr F(P)$.
Given $x \in H \setminus \{1_H\}$, we let $\mathsf L_H(x)$ be the set of all $...
- 10.5k
29
votes
1
answer
1k
views
Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 41.9k
2
votes
0
answers
50
views
Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...
- 617
2
votes
0
answers
169
views
What is the difference between a monosemiring and a semigroup?
What is the difference between a monosemiring and a semigroup?
The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity ...
- 203
1
vote
0
answers
29
views
Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (...
- 10.5k