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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Description of Green's relations $\mathcal{L}$, $\mathcal{R}$ in an inverse semigroup

Theorem: Let $S$ be an inverse semigroup, and let $x,y\in S$ and $e,f\in E_{S}$ then $x\mathcal{L}y$ if and only if $x^{-1}x=y^{-1}y$ $x\mathcal{R}y$ if and only if $xx^{-1}=yy^{-1},$ where $E_S$ ...
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Is there a topology on the full transformation semigroup?

$\mathscr T_X$ will denote the set of all functions from a non-empty set $X$ into itself, with the binary operation of composition $\circ$ making it a semigroup, called the full transformation ...
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$\mathcal{R}$-relation in full transformation semigroup

Let $T_{X}$ be the full transformation semigroup on $X$. For $\alpha$, $\beta \in T_{X}$ $$\alpha \mathcal{R}\beta \text { if and only if there exist }\gamma,\gamma' \in ...
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Do endormorphisms of a semigroup form a monoid? Automorphisms form a group?

I came across this theorem when studying thess lecture notes Theorem: Prove that The endomorphisms of a semigroup S form a monoid. The automorphisms of a semigroup S form a group. I do not ...
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230 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
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312 views

Ring without distributive law?

I recently came across a binary operation (in a very non-algebraic context - it's a way to organize a certain updating of log-likelihood-ratios) and was idly wondering whether it is any kind of ...
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Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
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1answer
140 views

Must a pseudoinverse of a von Neumann regular element be regular?

Background Let $R$ be a ring or a semigroup. We say that $x\in R$ is a von Neumann regular element of $R$ if there exists $y\in R$ such that $$xyx=x.$$ Any $y\in R$ satisfying the above equation is ...
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2answers
504 views

The compactness of the unit sphere in finite dimensional normed vector space

We define $ (\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $ \|.\|$ is defined to be any norm in $ \mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ ...
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If the product of two idempotents is idempotent, must the two idempotents commute?

It is a basic fact that when two idempotents $e,f$ in a semigroup $S$ commute, then $ef$ is an idempotent. Is the converse true? Is it true for idempotents in rings?
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Evan's theorem of embeddings into 2-generator semigroup.

I recently found this theorem when I am studying Semigroup presentations. Theorem: If S is a semigroup generated by denummerably many elements, then S can be embedded into a semigroup generated by ...
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1answer
120 views

Does a binoid contain an empty word?

$A$ is a finite alphabet. $A^*$ is the set of finite words or the free monoid generated by A. $A^w$ is the set of infinite words generated by A. Denote $A^\infty=A^*\bigcup A^w$. $X$ is a set of ...
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4answers
602 views

Cayley Graph of Semigroups

I did Google search and can't find a good answer. I thought I should ask experts here. http://en.wikipedia.org/wiki/Cayley_graph is for groups. My question is, Is there a special name for the ...
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3answers
356 views

Proving the set of functions from $\mathbb{R}$ to $\mathbb{R}$ form a semigroup under composition

Let $F=\{f\colon\mathbb{R}\to \mathbb{R}\}$. Show that $F$ is a semigroup under function composition. Is it a monoid? Is it a group? Is it Abelian? I know how to show that if $f,g,h$ are in $F$ ...
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213 views

Does every continuous time minimal Markov chain have the Feller property?

Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative offdiagonal entries). As explained for ...
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How many associative binary operations there are on a finite set?

I am studying Scott's book Group Theory. In the Exercise $1.1.17$ he asks us to show that if $S$ is a set and $|S|=n$, then there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$. But ...
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Eilenberg's rational hierarchy of nonrational automata & languages

In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised a Volume C dealing with "a hierarchy (called the rational ...
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Existence of universal enveloping inverse semigroup (similar to “Grothendieck group”)

Context In its simplest form, the Grothendieck group construction associates an abelian group to a commutative semigroup in a "universal way". Now I'm interested in the following nilpotent ...
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171 views

Group Theory Simple Proof

I am new at group theory, and I came across a question I would like help with Suppose we have a set $S$ with the only elements $p,q,r$. Let $a$ and $b$ be two elements of $S$. Consider the following ...
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clarification on the definition of meaningful product

I am studying Hungerford's book "Algebra". In the page 27 he defines the meaningful product as follows. Given any sequence of elements of a semigroup $G, > \{a_{1},a_{2},\dots\}$ define ...
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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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108 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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72 views

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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327 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
463 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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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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135 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\| + ...
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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 ...
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1answer
73 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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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 ...
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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 ...
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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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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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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
454 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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(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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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
115 views

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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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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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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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} = ...
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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 ...
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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 ...