Questions tagged [semigroups]

A semigroup is an algebraic structure consisting of a set together with a single 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/universal algebra. Please use the more specific tag (semigroup-of-operators) whenever appropriate.

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Well posed Vs Uniformly well posed Cauchy problem

Let $E$ be a Banach space, I am studying the following Cauchy problem for first ordre equations $$u'(t)=Au(t), \quad u(0)=u_0 \quad (t\geq0).$$ where $A:D(A)(\text{dense in $E$})\subseteq E\to E$ is a ...
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Is there a semigroup which admits no involution?

A semigroup with involution is a semigroup $(S;*)$ equipped with a unary function $f$ such that $f(f(x))=x$ and $f(x*y)=f(y)*f(x)$. I want to know, does there exist a semigroup for which there exists ...
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Good source to study the Laplace transform.

I am studying the theory of semigroups and its links with the spectral theory and the Laplace transform turns out to be the intermediary between the two. Any suggestions for good sources?
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When two semilinear morphisms are said to be equal?

Suppose $(S,A)$ and $(T,B)$ are two left semigroup acts. A pair of mappings $(\mu,f):(S,A) \to (T,B)$ is called a semilinear morphism if $\mu$ is a semigroup homomorphism and $f$ is a function ...
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2 votes
2 answers
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Do continuous conservative vector fields on $\mathbb{R}^3$ form a semigroup under composition?

Let $\mathcal{F}$ denote the set of conservative vector fields on $\mathbb{R}^3$ that are continuous. That is $$ \mathcal{F}=\{F:\mathbb{R}^3 \rightarrow \mathbb{R}^3: F \text{ is continuous and } F=\...
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2 answers
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Proving a set G is abelian group by only knowing * is associative with $a^2 \star b = b \star a^2$ for $a,b \in G$

I have a problem, given $G$ is a non empty set and we know that $\star$ is associative binary operation on $G$ such that $a^2 \star b = b \star a^2$ for all $a,b \in G$. I need to proof that $G$ is ...
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Does pointwise convergence imply measurability? and why?

$E$ is a Banach space and for every $u_0 \in E$ and $\{u_n\} \subset E$ with $u_n \to u_0$, we have $$S(t)u_n \to S(t)u_0$$ pointwise in $t \geq 0$. Why $t \mapsto S(t)u_0$ is measurable there ?
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Let $A$ be a semigroup and $X\subseteq A$. Define $X_0=X$ and $X_{n+1}=X_n\cup \{ab\, |\, a\in X,\, b\in X_n\}$. Then $\cup_{n=0}^{\infty}X_n=Sg^A(X)$

Let $A$ be a semigroup and $X\subseteq A$. Define $X_0=X$ and $X_{n+1}=X_n\cup \{ab\, |\, a\in X,\, b\in X_n\}$. Then $\bigcup\limits_{n=0}^{\infty}X_n=Sg^A(X)$, which is the subsemigroup of $A$ ...
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A finite semigroup has only trivial subgroups iff $\mathcal{H}$ is the identity relation.

I am struggling with the rightward implication. Here is some of my working: $(\Leftarrow)$ Every subgroup is contained within a maximal subgroup that is the $\mathcal{H}$ class of that subgroup's ...
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Why this integral vanishes for $t\leq 0$?

I am studying semigroup theory and in the proof of Hille-Yosida theorem, we have the integral $$\int_{\omega'-i\infty}^{\omega'+i\infty} \frac{e^{\lambda t}}{\lambda^3}R(\lambda;A)A^3u \, d\lambda$$ ...
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Proving that an associative binary operation gives rise to a group [duplicate]

I'm trying to prove the following claim. Let $S$ be a nonempty finite set, equipped with an associative operation $*: S \times S \to S$ such that, for every $x,y \in S$, there exists $z \in S$ such ...
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Numerical semigroup gerated by two elements

I have a question about numerical semigroups. It is known that if $a, b\in \mathbb{N}$ and $\gcd(a, b)=1$, then the numerical semigroup $\langle a, b \rangle$ has genus $\frac{(a-1)(b-1)}{2}$. My ...
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Group sufficiency conditions given an associative magma. [duplicate]

From Bourbaki's Algebra. I desire to see how others think about these interesting questions involving surjectivity and injectivity of translations. (4.2) This problem is marked as difficult. Problem a:...
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Can you, given any semigroup, define an identity element to make it a monoid

I'm wondering if I can "make up" an identity element, like so: I can define an element I such that any element x + I is equal to x, i.e.: I can redefine my set as [the old set] union with {I}...
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I have question about a semigroup for a Cauchy problem

please I have a question about semigroup theory, Let $(A, D(A))$ be the generator of a strongly continuous semigroup $(S(t))$ on $X$. Then the following Cauchy problem $$ \left\{\begin{array}{l} \dot{...
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2 votes
1 answer
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How to show that a compact semigroup for which the cancellation law holds is a compact group

Here is my problem: Set $G$ a compact semigroup (that is a Hausdorff compact space endowed with an associative continuous binary operation). Assume that the cancellation law holds i.e. for any $g,h,k \...
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Numerical semigroups generated by two elements

I have a question about numerical semigroups. It is known that if $a, b\in \mathbb{N}$ and $\gcd(a, b)=1$, then the numerical semigroup $\langle a, b \rangle$ has genus $\frac{(a-1)(b-1)}{2}$. My ...
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1 answer
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$S$ is the set of words generated by an alphabet. $A\subset S$ , $x\in S$. How to find if $x$ is generated by concatenating elements of $A$? [closed]

I am looking for references in relation to this problem. The name of the problem, texts or papers, efficient algorithms, and calculators. Example: $S$= the set of all finite binary strings, $A$= a ...
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Neumann & Dirichlet heat kernel resp. semi-group on compact manifolds

Given a smooth compact Riemannian manifold $(M,g)$ with boundary $\partial M$. Consider the initial & boundary value problem of the heat equation $$ \tfrac{\partial}{\partial t} u(x,t) = \Delta u(...
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A basic question about finite semigroup

Let $S$ be a finite semigroup. Recall that every element $a\in S$ determines a unique pair of positive integers $\iota=\mathrm{ind}(a)$ and $\rho=\mathrm{per}(a)$, called the index of $a$ and the ...
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3 votes
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Semilattice whose Subsets are All Closed -- does it have a special name?

Context: self-education. I am currently getting my head round semilattices. My understanding is that a semilattice $(S, \odot)$ is a semigroup whose operation $\odot$ is both commutative and ...
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the power of commuting

There is an interesting paper The power of commuting with finite sets of words. There I have a question about page 6 sentence:"...which is a contradiction because this word has no suffix ...
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Is there any category whose objects are semigroup acts and morphisms are semilinear morphism between acts?

I know that for a particular semigroup S all the S-acts form a category with S-act homomorphisms. My question is what happens if we do not fix the semigroup i.e. taking all semigroup acts (S,A) where ...
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2 votes
1 answer
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Prove that a monoid with property $ (a * a' = e) \lor (a' * a = e) $ is a group

Let $(G , *)$, $G \neq \emptyset$ be closed, associative, there is an identity element $ e $ in $ G $ and: $ (\forall a \in G)(\exists a' \in G) $ $ (a * a' = e) \lor (a' * a = e) $ Prove that G is a ...
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4 votes
1 answer
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Proof of a theorem about generators of uniformly continuous semigroup

I´m reading Vrabie and I'm have some problems understanding this proof. We have that $\{S(t); t\geq 0\}$ is uniformly continuous, so $\displaystyle\lim_{t\downarrow 0} S(t)=I$. Question 1: This ...
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Factors of Direct Composite of Subsemigroups are in fact Normal Subgroups

Seth Warner's "Modern Algebra" (1965), exercise 13.6. Context: self-study, from a many-years-ago maths degree which did not focus deeply on abstract algebra. If a group $G$ is the direct ...
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Idempotent Semigroup $S$ with Equivalence Relation $(a R b) \iff (aba=a), (bab=b)$: $S/R$ is commutative - why?

This is exercise 11.19 (h) in Seth Warner's "Modern Algebra". We are given that $S$ is an idempotent semigroup, that is: $\forall a \in S: aa = a$. Let $R$ be the equivalence relation ...
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Show that if $f$ is a homomorphism then the set of invertible elements $M^\times$ is commutative

I have to show the following. Let $M$ be a monoid. If $M \to M$, $f: a \mapsto a^2$ is a homomorphism, then $M^\times$ is commutative. So, if $f$ is a homomorphism, then for all $a,b \in M^\times$ we ...
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Index of an element in direct product of finite semigroups

Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the index of $(x,y)$ ? Is it equals $$ \max\{\mathrm{index}(x),\mathrm{index}(y)\}\ ? $$ My proof: If $\mathrm{index}(x)=i$, ...
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2 votes
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Period of an element in direct product of finite semigroups

Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the period of $(x,y)$ ? I know that if $$\mathrm{index}(x)=\mathrm{index}(y)$$, then the period of $(x,y)$ is $$\mathrm{lcm(...
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1 vote
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What is the group theoretical term to describe a uniquely factorizable semigroup?

I need to describe a particular characteristic of the set I'm working with, but I don't know the group-theoretical terms to describe it. Can someone help? The idea is that I have a semigroup, $G$, ...
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2 votes
1 answer
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Conditions for Group given Semigroup with Idempotent Element

Another exercise in Seth Warner's "Modern Algebra" (1965). This one is Exercise 7.15. Let $(S, \circ)$ be a semigroup. Let $(S, \circ)$ have an idempotent element $e$, that is, such that $e \...
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Hints on "Why is increasing monotonicity of growth rate of a function equivalent to convexity?

I have been studying Semi-groups theory and its application in Differential Equations and I have stumbled on this lemma: Let $I$ be an interval of $\mathbb{R}$ and $f: I\rightarrow\mathbb{R}$, $f$ is ...
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Left and Right Regular Representations are Permutations therefore Identity and Inverses Exist

Let $(S, \circ)$ be a semigroup. Let $a \in S$. It's a straightforward exercise to show that the left and right regular representations $\lambda_a$ and $\rho_a$ with respect to $a$ are permutations if ...
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1 vote
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How to prove that in a semigroup, two $\mathcal{L}$ classes in a $\mathcal{D}$ class are incomparable by $\leq_{\mathcal{L}}$ relation? [closed]

Let $S$ be a semigroup. I would like to prove that two $L$ classes (namely $L_1$ and $L_2$) such that both are in a $D$ class (that is $L_1\subseteq D$ and $L_2 \subseteq D$) are incomparable by $\...
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Left-elements of a numerical semigroup generated by two elements

A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
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Does semigroup and monoid have to be closed under the binary operation?

As stated in the title, I am wondering whether semigroup and monoid have to be closed under the binary operation. The reason I am asking about this is that in wiki pages of semigroup and monoid, the ...
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1 vote
1 answer
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Doubt in Location Lemma in Greens Relation Abstract Algebra!

I am unable to prove one part of rectangular lemma in green's relations. Let $S^1$ be a monoid. Then I need to prove that $m.m' \in D(m) \iff m.m'\in R(m) \cap L(m')$. How should I go about proving ...
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On the rank of finite semigroups

For any semigroup $S$, let $A$ be a non-empty subset of $S$. Then the subsemigroup generated by $A$ that is, the smallest subsemigroup of $S$ containing $A$, is denoted by $\langle A\rangle$. If there ...
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Is there general way to prove that for a given function exists formula with a constant amount of operations?

E.g there is a formula for a function that gives a number of idempotent functions for a set with finite size. The solution is $\sum_{k=1}^n{n\choose k}k^{n-k}$ but you need to sum intermediate results....
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Affine semigroup generating a lattice

Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$). Assume that $S$ generates $N$ as a group. Is it true that it ...
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Prop 1.7 One-parameter semigroups for linear evolution equations, Engel and Nagel.

Hello. Continuing with the study of the results of the book one-parameter semigroups for linear evolution equations by authors Engel and Nagel., I ran into some difficulties that I could answer but I ...
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3 votes
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Uniqueness of quotients in semigroup with divisibility?

Let $G$ be a semigroup, e.g., a set with an associative binary operation. Suppose further that $G$ has the divisibility property, e.g., for all $x,y\in G$ there exist $\ell,r\in G$ such that $\ell x=y$...
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1 vote
1 answer
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Deciding whether it is necessary to prove both directions in a proof

There are some exercises where I'm not fully sure that it's necessary to prove both directions. I'm using the below exercise as one example, though I'm not, per se, confused on how to solve this. This ...
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Neumann heat semigroup and first eigenvalue of $\Delta$

Let $\left(e^{t \Delta}\right)_{t \geqslant 0}$ be the Neumann heat semigroup in $\Omega$, and let $\lambda_{1}>0$ denote the first nonzero eigenvalue of $-\Delta$ in $\Omega$ under Neumann ...
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When do $2\times2$ matrices generate a finite semigroup?

Let $A_i$, $i=1,\ldots, k$, be $2\times2$ real-valued matrices with determinant 1 or -1. Under what circumstances is the semigroup generated by these matrices finite? I can see that this will be the ...
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5 votes
2 answers
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Proving $\nu_{t/n} \to \delta_0$ weakly as $n \to \infty$ for convolution semigroup

A convolution semigroup is a family of probability measures $(\nu_t)_{t \in I}$ on $\mathbb R^d$ with $I \subset [0,\infty)$ and $0 \in I$, for which $\nu_s * \nu_t = \nu_{s+t}$ for $s,t \in I$. I ...
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3 votes
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Vector spaces without additive inverses

I was writing out the axioms of a vector space, in preparation for teaching next week, and I started wondering: Do I actually need to impose that vectors have additive inverses? To be precise: Let $(F,...
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1 vote
1 answer
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hyperidentities, unknown evaluation

I do not follow the notion of hyper-identities very clearly. In the last line in the snippet, how looks the substitution $$x_1x_2x_1$$ for $F$, what is the result and can I see that this is not ...
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  • 3,578
1 vote
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Dynamical System Cocycles: (Semi)Group vs. Monoid Definitions - Examples When is Distinction Important?

Let $\mathbb{T}$ be a semigroup ("time") and $X$ a set ("state space") corresponding to a dynamical system (in particular $\mathbb{T}$ acts on $X$ in a known way, $(t , x) \mapsto ...
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