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Questions tagged [semigroups]

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.

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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 ...
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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 ...
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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 ...
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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)...
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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)\...
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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}\...
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Convolution with an analytic semigroup

Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution $$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$ ...
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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 ...
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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 ...
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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 ...
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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\...
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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: ...
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1answer
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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}...
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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 $C_0$-...
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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 ...
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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 ...
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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 ...
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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-...
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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 $W(\Sigma; v)$. ...
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1answer
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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 ...
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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 ...
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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 $X$ ...
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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)...
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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)=\{...
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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 (...
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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 ...
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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 ...
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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 \...
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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 \...
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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 ...
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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 $M_2$, $M=M_1\oplus M_2$...
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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)...
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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 ...
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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=\...
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1answer
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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(...
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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 ...
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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^{-...
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1answer
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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{...
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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 $G$ ...
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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 ...
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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 ...
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1answer
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Generator of Wiener process and its running maximum

This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty). See https://math.stackexchange.com/questions/1274775/generator-of-wiener-process-and-its-...
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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\...
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How can I have a copy of seminary note of Prof. Barbu

I would like to refer this luminous seminary note by Prof Viorel Barbu: ``Seminar on nonlinear semigroups and evolution equations'' published in 1970, iasi. There seems to be no libraries holding ...
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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 ...
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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,...
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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 ...
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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 ...
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Do we have that $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ are equivalent norms?

Is it possible that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms? This results is pretty easy and straightforward for $p=2$ ...
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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 ...