Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [semigroups-and-monoids]

A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). Of course, any monoid or group is also a semigroup.

1
vote
0answers
47 views

Schemes for conditional distributions (monads)

(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.) Suppose you have a monad ...
-3
votes
1answer
181 views

A common name for a functorial construction of Commutative Algebra?

I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name. Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
1
vote
0answers
77 views

When a semigroup ideal is a determinantal ideal?

Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
4
votes
0answers
71 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 ...
2
votes
1answer
111 views

bp continuity of Markov operators / semigroups

Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
1
vote
0answers
36 views

Examples of associative inducers and other inducers

I am curious about how well the following technique can produce algebraic structures and semigroups in particular. Let $(X,\circ)$ be a semigroup. Let $Y$ be a set and let $L:X\rightarrow P(Y)$ be a ...
2
votes
1answer
68 views

Well-posedness of wave equations with time-dependent coefficient

Let us consider the following wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \\ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\ y(0,x)...
2
votes
1answer
128 views

Bounded solution for parabolic equation

Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\...
1
vote
0answers
96 views

Has the “semidirect monoid of a semiring” been considered anywhere?

Given a semiring $S$, we get a monoid $M(S)$ as follows: The underlying set of $S$ is $S^2$ The identity element is $(0,1)$ The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
1
vote
0answers
39 views

Interpolation theory

Consider the interpolation space $Z=(X,Y)_{\theta,p}$, in the case $Y\subseteq X$ do we have that the following norm: $x\longrightarrow\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\...
1
vote
0answers
41 views

Generating totally ordered free commutative monoids

Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$. When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am ...
1
vote
1answer
56 views

Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations

It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...
2
votes
0answers
76 views

How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process?

In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ...
1
vote
1answer
90 views

Do these sorts of submonoids go by a particular name?

Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows: $$r(x)=\{y\in M:xy=x\}$$ $$l(x)=\{y\in M:yx=x\}$$ Do these sorts of sub-monoids go by a particular name?...
7
votes
1answer
466 views

Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?

Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
7
votes
1answer
312 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\...
0
votes
0answers
76 views

Bounded and sectorial operators

Is there any assumption for a bounded operator to be sectorial ? Is there any characterization of such operators ? Here, the definition of sectorial operators follows the book of Markus Haase: ...
2
votes
1answer
72 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
votes
1answer
82 views

Generation of strictly contraction Semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...
1
vote
0answers
62 views

Using semigroup theory for nonautonomous semilinear equations

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...
6
votes
1answer
364 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 ...
6
votes
1answer
237 views

Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$ is not ...
7
votes
1answer
136 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 ...
4
votes
0answers
58 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 ...
3
votes
0answers
46 views

A question about a theorem in 'Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators'

I have asked this question on stack and someone advised me to ask it here. The link is https://math.stackexchange.com/questions/2900658/a-question-about-a-theorem-in-quantum-dynamical-semigroups-...
3
votes
1answer
96 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
1answer
192 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 ...
13
votes
2answers
247 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 ...
0
votes
0answers
68 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
1
vote
0answers
95 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)...
0
votes
0answers
195 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)=\{...
3
votes
1answer
54 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 $\sigma\subseteq V$...
0
votes
0answers
27 views

Volterra equation of the first kind with an exponential kernel

I am interested in an approximation $\hat{f}$ as well as the error estimate $\|\hat{f} - f\|_{L^2}$ for $f\in L^2([0,T];\mathbb{R})$ in the following Volterra equation $$Af(t) = \int_0^t e^{-\lambda (...
6
votes
0answers
279 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 ...
3
votes
0answers
29 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 ...
0
votes
0answers
38 views

On the relation of ideals and $\mathcal J$-classes in semigroups

Given a semigroup $S$, a subset $I$ is called an ideal iff for every $s \in S$ we have $sI, Is \subseteq I$. Further we set $$ s \le_{\mathcal J} t :\Leftrightarrow SsS \cup \{s\} \subseteq StS \cup \...
1
vote
1answer
70 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 Froebenius number of $S$, $g(S)=|\mathbb N \...
1
vote
1answer
90 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 ...
8
votes
2answers
295 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: ...
8
votes
1answer
171 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
vote
1answer
90 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 $M_2$, $M=M_1\oplus M_2$...
6
votes
1answer
181 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
votes
0answers
58 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)...
5
votes
0answers
252 views

Compactness of semigroups of one-dimensional diffusions

I have a question about semigroups of one-dimensional diffusions. Let $X$ be the Ornstein Uhlenbeck process on $\mathbb{R}$. The generator is expresses as $$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$ It is ...
-4
votes
1answer
193 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 ...
1
vote
0answers
43 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 ...
9
votes
1answer
211 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. ...
0
votes
0answers
81 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=\...
1
vote
1answer
92 views

Separation property for non-injective flows

Let us consider a non-injective flow $X$ on $\mathbb{R}^d$, i.e. a continuous map $X:\mathbb{R}_+\times \mathbb{R}^d \to \mathbb{R}^d$ with $X(0,\cdot)=id$ and satisfying the semigroup property $X(t,X(...
3
votes
1answer
88 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 ...