A semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup with an identity element is called monoid. This tag is most frequently used for questions related to the concept of semigroups in the context of abstract algebra. Please use the more ...

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Left and Right Inverses with semigroups

Having the semigroup $(F,\circ)$ where $F=\{f: f: \mathbb{N}\to \mathbb{N}, \mathrm{Dom}(f) = \mathbb{N}\}$. The identity $e∈F$ is the function $e(n) = n$, define the function $g(n) = m$ if ...
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125 views

Proving identity of a $C_0$ semigroup

We have a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$ on a Banach space X with a generator $ A:D(A)\subset X\to X$. I need to show that $\displaystyle T(t)x = \sum\limits_{n=0}^{\infty} ...
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165 views

Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue

Consider the heat equation defined as $$\begin{cases} \partial_{t}u(x,t)=\Delta_{x}u(x,t) &\mbox{ } t>0\mbox{, } u\in U \\ u(x,t)=0 &\mbox{ } t\ge 0\mbox{, } x\in\partial U \\ u(x,0)=g(x) ...
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126 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
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147 views

Proving that T(t)x is in the domain

$(T(t))_{t\ge 0}$ is a $C_{0}$-semigroup on a Banach space $X$ with generator $A:D(A)\subset X\to X$. For $k\ge 2$, define $$D(A^{k}):=\{x\in D(A^{k-1})|A^{k-1}x\in D(A)\}$$ I want to show that for ...
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39 views

A criterion for invertibility of a bounded linear operator.

I'm studying Semigroup Theory and I wasn't able to understand a step in a certain proof. As far as I have been able to understand, the author used the following result: If $A$ is a bounded linear ...
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1answer
55 views

Some weaker axiom than “no nontrivial zero divisors.”

I would like to know if there a standard term for or well-known applications of the following axiom for rings or semigroups with zero (which is weaker than the "no nontrivial zero divisors" axiom): ...
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35 views

What is the algebraic structure of sets with a “splitting” operation?

I noticed that any convex shape can be split along a straight line and produce two shapes that are also convex. The general pattern seems like the dual of a semigroup—what is it called? A cosemigroup? ...
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75 views

Domain of the infinitesimal generator of the shift semigroup

In a course at my university, we study strongly continuous semigroups and their infinitesimal generators. In a simple example, we take a look at a shift semigroup. let $ T $ be an operator on $ X = ...
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56 views

Invariance of semigroups

$A$ is the infinitesimal generator of the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ and $V$ is a one dimensional linear subspace of $X$. I want to show that $V$ is $T(t)$-invariant $\iff$ ...
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43 views

Proving strong stability of semigroup

$X$ is the Hilbert space $L^{2}(0,\infty)$ and let $T(t):X\to X$ with $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$. I want to prove that the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ is strongly ...
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114 views

Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ ...
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1answer
31 views

The intersection of *-semigroups with I-semigroups is the class of inverse semigroups?

Howie in his Fundamentals of Semigroup Theory, 2nd ed., p. 103 writes The class of U-semigroups for which the unary operation satisfies the conditions both for a *-semigroup and for an I-semigroup ...
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Cayley's Theorem for Semigroups

I've read and fully understand Cayley's theorem for groups, however when I get to the theorem for semigroups I come to a complete stop. I've figured that the identity and cancellative properties are ...
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1answer
28 views

An easy example of a non-seminormal (commutative) monoid in generators and relations?

Let $M$ be a commutative monoid. We denote by $M^{Gr}$ its Grothendieck Group (i.e. group of fractions). We then have a morphism $q:M\rightarrow M^{Gr}$. We say that a commutative monoid $M$ is ...
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100 views

A rather abstract strongly continuous semigroup

Define $X$ as the Hilbert space $L^{2}(0,\infty)$ and let the operators $T(t):X\to X$, $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$ I want to show that $(T(t))_{t\ge 0}$ is a ...
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30 views

Is there a logarithmic size generating set for some classes of finite semigroups?

Following my question Why is the minimum size of a generating set for a finite group at most $\log_2 n$?, we know that finite groups have generating sets of size at most $\log_2 n$, and a similar ...
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63 views

Does the “equality semigroup” have an accepted name?

Given a set $G$, we get a semigroup on $G \cup \{0\}$ as follows: Define $x^2 = x$ for all $x \in G \cup \{0\}$. Define $xy = 0$ for all distinct $x,y \in G \cup \{0\}$. Question 0. Does this ...
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53 views

Find all homomorphisms from a semigroup $(\mathbb{Z},+) \to (\mathbb{Q},+)$

In the book Karpfinger & Meyberg (2013) "Algebra" I encountered this problem. So far I have figured out $\tau(x) = qx$ wehre $q \in \mathbb{Q}$. Are there other homomorphisms? Is there a general ...
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224 views

Proof involving strongly continuous semigroups.

Let $ (T(t))_{t \geq 0} $ be a $ C_{0} $-semigroup on a Hilbert space $ X $ with an infinitesimal generator $ A $, and let $ \rho \in (0,1) $. I want to prove that $ \displaystyle \sup_{t \geq 0} \| ...
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46 views

Showing that A is NOT an infinitesimal generator

As a state space, choose $X=L^{2}(0,1)$. Let $A$ be defined as $\displaystyle Af=\frac{df}{d\zeta}$ with domain $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous and $\frac{df}{d\zeta}\in ...
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Finding the infinitesimal generation of a strongly continuous semigroup

Let $X$ be a Hilbert space, $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I have already shown that $T(t)$ defines a $C_{0}$ semigroup. But now I need ...
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31 views

Proving one of the properties of a strongly continuous semigroup

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I want to show that $T(t+\tau)=T(t)T(\tau)$. So we have that ...
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31 views

semigroups defined on banach algebra

Let $(X,\|.\|)$ be a Banach space and $Z:=\{Z(t)\}_{t\geq 0}$ is strongly continuous semigroup defined on it. If $X$ turns out to be a Banach Algebra, i.e. for $x,y\in X$, $xy\in X$. Is $Z$ still ...
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52 views

Does this show that it is a bounded linear operator?

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$. I want to show that $\displaystyle e^{At}:=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}=T(t)$ defines a strongly continuous semigroup (i.e. a ...
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1answer
234 views

L,R,H,D,J relations on a completely simple semi group represented my a rees matrix

I am trying to tackle the following semigroup question. I can't see why my answer is wrong but I haven't used the fact the semigroup is COMPLETELY simple anywhere so I think there must be an error ...
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205 views

Proof of Lallement’s Lemma

i am a bit weak with my congruence manipulation when it comes to semigroup theory. Could you possibly check my proof and give me constructive criticism on it? Lemma: Let $S$ be a regular semigroup ...
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Is the unit ball of $H^\infty(\mathbb{D})$ a metrizable topological semigroup under multiplication?

The space $H^\infty(\mathbb{D})$ of all bounded holomorphic functions on the open unit disc carries many different topologies. One such topology is given by uniform convergence on compact subsets; ...
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1answer
173 views

Drawing a directed graph in Maple

I have been playing around with some Maple today for a semigroup/ graph theory style project. I want to draw a left cayley graph with vertices $\{1,....,12\}$ and edges $[5, 5]$, $\{[6, 2].[7, 1], [1, ...
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Prove that $\{S(t)\}_{t \ge 0}$ is not a contraction semigroup on $L^\infty(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the ...
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Contraction semigroup on $X$ with generator $A$

Let $\{S(t)\}_{t \ge 0}$ be a contraction semigroup on $X$, with generator $A$. Inductively define $D(A^k):=\{u\in D(A^{k-1}) \mid A^{k-1}u \in D(A)\}$ ($k=2,\ldots$). Show that if $u \in D(A^k)$ ...
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“Universal solution” to factorization of arithmetic mean through a semigroup homomorphism.

Headnote A: my apologies for the long winded exposition; I couldn't find a more compact way to convey what I'm looking for, why I'm looking for it, and what have I tried myself. Suggestions for ...
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59 views

Justifying an equality involving a closed operator $A$

Justify the equality $$A \int_0^\infty e^{-\lambda t} S(t) u \, dt = \int_0^\infty e^{-\lambda t} AS(t) u \, dt$$ used in (16) of §7.4.1. (Hint: Approximate the integral by a Riemann sum and recall ...
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96 views

Prove that $\{S(t)\}_{t \ge 0}$ is a contration semigroup on $L^2(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the ...
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134 views

Properties of resolvent operators

I am asked to prove the identities of $(12)$ and $(13)$, which are given on page 438 of the textbook PDE Evans, 2nd edition as follows: THEOREM 3 (Properties of resolvent operators). (i) If ...
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Doubt about semigroups in this article (anyone can help).

I need help in this article. My doubt is very arithmetical and I think follows directly from the definitions. So I think anyone could help me. The author defines what is a semigroup, gaps and ...
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32 views

Invertibility of operators related to Markov processes in Ethier-Kurtz

Lemma 2.3 of the book by Ethier and Kurtz (first edition, I believe) defines $$ g_n := (\lambda - A)(\lambda_n - A)^{-1}g $$ for some fixed $ g $ but I see no guarantee that $(\lambda_n - A)^{-1} g ...
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64 views

A finite pseudo-ring such that $ab^2=b$ for some $(a,b)\in A$

I have this exercise which I found very difficult: $(A,+,.)$ is a finite pseudo-ring (a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity), and ...
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Semi group presentation $<a, b | a^{2} = b^{2} = 0, aba = a, bab = b>$

Another semi group question here, trying to get my head around the topic. Consider the semi group $S=\left<a, b | a^{2} = b^{2} = 0, aba = a, bab = b\right>$ I need to prove that $S$ has order ...
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185 views

If S is finite and $x/\rho$ is an idempotent of $S/\rho$ then $x/ \rho$ contains an idempotent

Currently working on the following problem, need a little help with the solution to the last part, any hints? Q: Let S be a semigroup, and let ρ be a congruence on S. Prove that if e ∈ S is an ...
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50 views

How is the following expresson be obtained and the meaning of the expression in blue box?

Let me introduce the term {$E_\lambda:\lambda \geq0$} is the spectral resolution of identity of a self adjoint densely defined, positive and closed operator $A:D(A)\subset X\rightarrow X$ , Where X ...
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1answer
60 views

A semigroup with identity having exactly one idempotent is a group

Let $G$ be a finite semi-group with identity such that it has only one idempotent.Is $G$ a group? It only remains to show that for any $a\in G$ $\exists b\in G$ such that $ab=ba=e$ where $e$ is the ...
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48 views

Isomorphism of a semigroup S and (Aᴬ, ⋅).

I would like to ask you for help with proving the following theorem from our textbook: Any semigroup is isomorphic to a subsemigroup of ($A^A, \cdot$) for a suitable set A. The theorem is then ...
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138 views

Associativity in category theory [closed]

In Category Theory, the composition of morphisms must be associative. What would happen if we give up this associative law? For example Lie algebras are vector spaces with non-associative binary ...
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Shift operators and c0 semigroups

I am asked where the bounded continuous functions on $\mathbb{R}$ (with sup norm) with the right shift operators ( $T_t(f)(x)=f(x-t)$ ) form a c0 semigroup. I believe the answer is no, but I am ...
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1answer
28 views

Closed linear operator

I am having some trouble in showing the following map is closed: For $f\in L^2(\mathbb{R^2})$ with $(x+iy)f(x,y)\in L^2(\mathbb{R^2})$, $M(f)(x,y)=(x+iy)f(x,y)$. I am also asked to find the ...
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108 views

Find a first order sentence satisfied in every finite semigroup but not in every compact semigroup

Several properties that hold in nonempty finite semigroups also hold in nonempty compact semigroups. Furthermore, many of these properties can be formulated by a first order sentence. For instance, ...
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187 views

Semigroups and solutions of equation

It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ya=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ya=b$ has ...
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59 views

Semigroup satisfying the cancellation property which cannot be embedded in a group [duplicate]

There is a basic result about commutative semigroups which says Any commutative semigroup satisfying the cancellation property can be embedded in a group. However, I read that Not every ...
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58 views

Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...