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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Equivalent Norms for Intermediate Subspaces

Let $(X,\left\|\cdot\right\|)$ be a Banach space, and let $\left\{T(t) : t\geq 0\right\}$ be an equibounded strongly continuous semi-group on $X$. Define a functional ...
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Is my proof of the following correct?

Question: Let $M,P,Q$ be semigroups and $\sigma:M\rightarrow P$, $\rho:M \rightarrow Q$ be morphisms with $\sigma$ surjective. Then $\ker \sigma \subseteq \ker \rho$ if and only if $\rho$ factors ...
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spectrum of weighted translation semigroup

My Banach space is $\mathcal X=\rm{L}^1(\mathbb R_+)$. I would like to know the spectrum of $A\phi(x)=-\phi'(x)-f(x)\phi(x)$ on $D(A) = \{g\in\mathcal X,\ g\text{ absolutely continuous}, g(0)=0\text{ ...
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Are there any resources on this notion of a “directed” semigroup?

Given two additive semigroups $G$ and $H$, and a semigroup action $ \& : G \times H \to G$, then intuitively $H$ can be thought of as a set of "directions" in which we can move in the space $G$. ...
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1answer
55 views

Free idempotent semigroup with 3 generators

On Finite free objects I asked examples of finite free objects. I got answer "The free idempotent semigroup satisfies $x^2=x$ for each element $x$. And I confirm it is finite if it is finitely ...
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1answer
62 views

Computing the shortest encoding for a transformation

Here goes... Let $n = 2^p$ for some $p \in \mathbb{N}$. Let $\mathcal{T}_n$ denote the set of all transformations on $\mathbb{Z}_n$ (viz. the transformation monoid). Pick $C$ to be an order ...
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Are there any other useful semigroups aside from $e^{At}$

$e^{At}$ trivially satisfies both properties of semigroup namely $T(t+s) = T(t)T(s)$ $T(0) = I$ Does there exist any other commonly used operators aside from $e^{At}$ that is a semigroup? ...
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1answer
31 views

Build a Markov process from a transition semigroup

Suppose that we have a Markov process $X$ with time-homogeneus transition probability function given by $p(t,x,A)$, with $t>0, x\in E, A \in \mathcal E$, where $(E,\mathcal E)$ is the state space. ...
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Order of calculation in a Semigroup.

If $(A,\cdot)$ is a semigroup, i.e. if we have: $\forall (x,y,z)\in A^{3}, (x\cdot y)\cdot z=x\cdot(y\cdot z)$, then the order of calculations doesn't matter no matter the number of factors and we can ...
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1answer
62 views

Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very ...
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Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
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Let $G$ be a semigroup. If for any $a,b\in G$, the equations $ax=b$ and $ya=b$ are solvable, then $G$ is a group.

I at a loss here. Showing that $G$ has an identity is the difficult part here. Obviously for each $a\in G$ there are $b,c\in G$ such that $ab=a$ and $ca=a$, but I need a hint as to how to prove these ...
4
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1answer
39 views

The identity elements of two multiplication satisfying interchange law.

Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 ...
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3answers
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A semigroup with three or four elements without identity [duplicate]

Does there exist a semigroup with three or four (or finite) elements, without identity? I tried to construct such an example, but every example I tried to construct had an identity element.
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What is the right (a good) definition for a dual monoid?

Suppose we have the free abelian monoid $S = \{a^m : m \in \mathbb{N}_0\}$ on the set of one element $X = \{a\}$. The binary operation on the monoid is denoted by $\cdot$. If $(T,\ast)$ is another ...
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Ellis semigroup

I have only seen the construction of the Ellis semigroup applied to a group endowed with the discrete topology. Does the same idea works for any topological group? If not, where does it go wrong? (I ...
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1answer
47 views

Question about kinda-sorta identities in algebra

A friend showed me some semi-groups which were constructed as follows. Let $Q = \{ 1, \ldots, n \}$ be a finite set, and let the operation $\cdot$ be defined on $(Q^{2})^{2}$ by \begin{align*} ...
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23 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
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57 views

Classical solution involving semigroups

Let $\{T_{D}(t)\}_{t\ge 0}$ a $C_{0}$-semigroup with generator $A+D$, with $D\in\mathcal{L}(X)$ and let $x_{0}\in \mathcal{D}(A)$. I want to show that ...
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1answer
20 views

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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1answer
117 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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131 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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1answer
98 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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1answer
146 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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2answers
31 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
51 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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1answer
32 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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Lummer-Phillips theorem for generator of strongly continuous semigroup

Definition: Let $P_{1}\in\mathbb{K}^{n\times n}$ be invertible and self-adjoint, let $P_{0}\in\mathbb{K}^{n\times n}$ be skew-adjoint, i.e., $P^{\ast}_{0}=-P_{0}$, and let $\mathcal{H}\in ...
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1answer
33 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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1answer
35 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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1answer
36 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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Bounding the operator

If $X$ is the Hilbert space $L^2(0,\infty)$ equipped with the inner product $$\langle f,g\rangle :=\int_0^\infty f(\zeta)\overline{g(\zeta)}(e^{-\zeta}+1) \, d\zeta,$$ and the operator $T(t):X\to ...
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1answer
102 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
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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
20 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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1answer
90 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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1answer
24 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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1answer
56 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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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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Perturbations of Positive Semigroups with Applications

P is an operator, if P is p-admissible for A then P is p-admissible for A + P. the proof of this theorem is based on the theorem Miyadira-Voiyt. but if we will demonstrate this result in general I ...
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1answer
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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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1answer
32 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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37 views

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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Showing that an operator generates a unitary group

Consider the following operator on $X=L^{2}(0,1)$: $\displaystyle Af=\frac{df}{d\zeta}$ with domain: $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous, $\frac{df}{d\zeta}\in L^{2}(0,1)$ and ...
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1answer
22 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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25 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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39 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
210 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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1answer
168 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 ...