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 ...

learn more… | top users | synonyms (1)

3
votes
1answer
298 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 ...
10
votes
1answer
163 views

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
votes
1answer
51 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 c)\circ_1(b\...
0
votes
3answers
73 views

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.
1
vote
0answers
60 views

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 ...
1
vote
1answer
48 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*} (t(\...
1
vote
0answers
26 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 ...
0
votes
1answer
23 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 $5(m-1)&...
2
votes
1answer
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} \frac{...
0
votes
1answer
168 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) &...
2
votes
1answer
130 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 $D(A)...
1
vote
1answer
149 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 ...
2
votes
2answers
41 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 ...
2
votes
1answer
56 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): ...
0
votes
1answer
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?
1
vote
1answer
77 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 = L^...
0
votes
1answer
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 $\iff$ $V=span{\{\phi\}}$...
1
vote
1answer
44 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 ...
1
vote
1answer
116 views

Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ $\displaystyle\frac{\...
1
vote
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 ...
3
votes
2answers
110 views

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 ...
1
vote
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 ...
1
vote
1answer
102 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 $C_{0}$-...
2
votes
1answer
31 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 ...
4
votes
1answer
64 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 ...
1
vote
2answers
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 ...
5
votes
1answer
226 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} \| ...
0
votes
1answer
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 L^{2}(0,...
0
votes
2answers
59 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 ...
0
votes
1answer
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 $\...
0
votes
0answers
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 ...
1
vote
1answer
53 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 $C_{0}$-...
1
vote
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 ...
0
votes
1answer
207 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 ...
2
votes
0answers
18 views

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; ...
1
vote
1answer
179 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, ...
4
votes
1answer
142 views

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 heat ...
2
votes
1answer
87 views

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 $u \in D(A^k)$ for ...
1
vote
0answers
20 views

“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 ...
4
votes
1answer
69 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 $A$...
0
votes
1answer
102 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 heat ...
2
votes
2answers
141 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 $\...
2
votes
0answers
30 views

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 non-...
0
votes
1answer
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 ...
2
votes
2answers
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 $(...
1
vote
2answers
103 views

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 ...
1
vote
1answer
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 ...
2
votes
1answer
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 be ...
2
votes
1answer
61 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 ...
0
votes
2answers
49 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 ...