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.

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Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
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3answers
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What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
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Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product A*B={PQ|P in A & Q in B & dim(Q)=codim(P)}, and ...
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1answer
109 views

Necessary and sufficient condition of being dissipative

I want to know a necessary and sufficient condition on $m:\Omega \mapsto\mathbb{C}$ such that the multiplication operator $M_{m}$ is dissipative in $L^{p}(\Omega)$, where $\Omega$ is a Banach space. ...
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0answers
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In a semigroup $S=\{a_1, \ldots, a_n\}$, any product $a_1 * \ldots * a_i, 1 \le i \le n$ is unique [duplicate]

Possible Duplicate: How does one actually show from associativity that one can drop parentheses? Claim: Let $(S, *)$ be a semigroup and $a_1, \ldots, a_n \in S$. Then $a_1 * \ldots * a_n$ ...
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2answers
365 views

A cyclic subsemigroup of a semigroup S that is a group

I came across this problem while reading some lectures about semigroups here Lecture Notes on Semigroups by Tero Harju. Page $10$. He named it a nontrivial exercise. Let $r$ be the index and $p$ the ...
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1answer
54 views

$A$ is a subsemigroup of $S$ if and only if $A^{2}\subset A$

Question: Show that $A$ is a subsemigroup of $S$ if and only if $A^{2}\subset A$. The subset here may not necessarily be proper. My approach, Suppose $A$ is a subsemigroup of $S$, then for ...
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3answers
511 views

Cayley tables for semigroups of order $\le 8$

I need Cayley Tables for semigroups of order $\le 8$. If someone knows where can I find this information, please let me know. I know that this information is stored in GAP(Groups, Algorithms, ...
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1answer
259 views

When a semigroup can be embedded into a group

Under what assumptions can a semigroup $(S,*)$ be embedded into a group?
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1answer
139 views

how to prove the generator of semigroup is a Banach space

I am not familiar with semigroup theory, so please stand with my dummy question. Say, $A$ is the generator of a semigroup, consider space $X_{n} = D(A^{n})$ with graph norm, $\|f\|_{A^{n}}:=\|f\| + ...
17
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2answers
658 views

How to get a group from a semigroup

I am sorry if my question is too simple. Is every semigroup associated to a group? If no, what conditions should be satisfied for a semigroup to have an associated group? If yes, how can I find the ...
4
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1answer
74 views

Nomenclature for distributive action between semigroups

Let $(G,*)$ and $(H,+)$ be semigroups. Let $\cdot$ be an action of $G$ on $H$, such that $\cdot$ distributes over $*$. [I.e., $(g_1 * g_2) \cdot h = g_1*(g_2\cdot h)$, and $g\cdot(h_1+h_2) = (g\cdot ...
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1answer
126 views

Subsemigroup generated by an element contains unique idempotent [duplicate]

Possible Duplicate: A cyclic subsemigroup of a semigroup S that is a group My homework: An element $s^{i+k}$ on the cycle is idempotent iff $$ s^{i+k} = s^{2i+2k} ,$$ or equivalently ...
5
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0answers
254 views

Inverse of the Grothendieck group construction?

Is there an "inverse" of the Grothendieck group construction that would generate a (somehow "simplest") Abelian semigroup given some (suitably qualified, if necessary) Abelian group? (I realize that ...
6
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1answer
705 views

Why is the free monoid free?

I have been reading up on monoids recently and came across the free monoid Σ*, which (if I understand correctly) is an initial object in the category of monoids, meaning that there is a ...
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3answers
217 views

Is a contraction semigroup infinitesimal operator bounded?

Let $T_t:L\to L$ be a semigroup of linear operators $T_t$ acting on a Banach space $L$. Assume that $$ \|T_t\| := \sup\limits_{f\in L}\frac{\|T_tf\|_L}{\|f\|_L} \leq 1 $$ for all $t\geq 0$. The ...
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1answer
298 views

Proof that $\Delta$ generates analytic semigroup

First off, I apologize for asking a question which I'm sure has been studied to death, but I can't seem to find an answer with google. I want to see a proof that the Laplace operator $\Delta$ with ...
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1answer
520 views

What does Trotter Product Formula mean?

For some reason, I have to work with Trotter product formula recently, but I do not have a strong background in functional analysis. The following is the statement of the formula from MathWorld ...
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2answers
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(Organic) Chemistry for Mathematicians

Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do ...
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2answers
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Continuity of semigroups on $L^2$ and $L^1$: Is this simple proof correct?

Let $(X, \mu)$ be a $\sigma$-finite measure space, and $P_t$ a symmetric, Markovian, strongly continuous contraction semigroup on $L^2(X,\mu)$. (Markovian means that if $f \in L^2$ with $0 \le f \le ...
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1answer
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Is there a special name for a semigroup whose multiplication is a constant function?

Let $S$ be a (commutative) semigroup with distinguished element 0 such that $ab=0$ for $a,b\in S.$ Of course this is a very simple family of semigroups, defined only by their cardinality. Does it ...
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1answer
78 views

Linear operators and Markov semigroups

I was trying to understand the Ergodic theory recently, but I don't really have any knowledge about linear operators, Markov semigroups etc. so I didn't even fully understand the definition. Could ...
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1answer
173 views

is there an analogue of short exact sequences for semigroups?

Since semigroups don't need to have an identity element, I was wondering if there's any kind of short exact sequence for semigroups.
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1answer
563 views

Uniqueness mild solution of $\dot{x} = A x$

Let $A$ be the infinitesimal generator of a $C_0$-semigroup $(S(t))_{t \geq 0}$. Now, for every $x_0 \in X$ the map $t \mapsto S(t) x_0$ is a mild solution of \begin{equation}\label{eq:1} \dot{x} = ...
4
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1answer
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Semigroup with “transitive” operation is a group?

I have a semigroup $G$ (a set with associative binary operation) such that for all $a,b\in G$ there exists $x,y\in G$ such that $ax=ya=b$. Is this property enough to show that $G$ is a group, and if ...
25
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9answers
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Are there any interesting semigroups that aren't monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there any ...