# 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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### 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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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 $\... • 106 0 votes 0 answers 40 views ### Left-elements of a numerical semigroup generated by two elements A numerical semigroup$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$can be expressed as a ... • 125 1 vote 2 answers 43 views ### Does semigroup and monoid have to be closed under the binary operation? As stated in the title, I am wondering whether semigroup and monoid have to be closed under the binary operation. The reason I am asking about this is that in wiki pages of semigroup and monoid, the ... • 397 1 vote 1 answer 56 views ### Doubt in Location Lemma in Greens Relation Abstract Algebra! I am unable to prove one part of rectangular lemma in green's relations. Let$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')$. How should I go about proving ... • 106 0 votes 0 answers 22 views ### On the rank of finite semigroups For any semigroup$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$. If there ... • 1,107 0 votes 0 answers 85 views ### Is there general way to prove that for a given function exists formula with a constant amount of operations? E.g there is a formula for a function that gives a number of idempotent functions for a set with finite size. The solution is$\sum_{k=1}^n{n\choose k}k^{n-k}$but you need to sum intermediate results.... 2 votes 0 answers 50 views ### Affine semigroup generating a lattice Everything is assumed to be finite-dimensional. Let$S$be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice$N$). Assume that$S$generates$N$as a group. Is it true that it ... 0 votes 0 answers 11 views ### Prop 1.7 One-parameter semigroups for linear evolution equations, Engel and Nagel. Hello. Continuing with the study of the results of the book one-parameter semigroups for linear evolution equations by authors Engel and Nagel., I ran into some difficulties that I could answer but I ... • 2,509 3 votes 1 answer 40 views ### Uniqueness of quotients in semigroup with divisibility? Let$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$... • 4,878 1 vote 1 answer 38 views ### Deciding whether it is necessary to prove both directions in a proof There are some exercises where I'm not fully sure that it's necessary to prove both directions. I'm using the below exercise as one example, though I'm not, per se, confused on how to solve this. This ... • 1,184 0 votes 0 answers 41 views ### Neumann heat semigroup and first eigenvalue of$\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 ... 2 votes 1 answer 65 views ### 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 ... • 99 5 votes 2 answers 94 views ### 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 ... • 3,001 3 votes 2 answers 101 views ### 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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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 ...
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 ...