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 ...
26
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3answers
941 views

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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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 ...
17
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2answers
630 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 ...
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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 ...
14
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1answer
416 views

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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5answers
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Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...
12
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1answer
319 views

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 ...
11
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669 views

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 ...
9
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1answer
266 views

Properties of Group $\to$ Monoid $\to$ Semigroup.

In How to get a group from a semigroup Arturo Madigin writes: So if we look at the composition of the forgetful functors $\bf Group \to \bf Monoid \to \bf Semigroup$, we obtain a right adjoint by ...
9
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1answer
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Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...
9
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1answer
101 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
9
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0answers
210 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
8
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3answers
266 views

give a counterexample of monoid

$G$ is a monoid,$e$ is its identity,if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$. if not,please prove $b=c$ thanks a lot.
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1answer
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Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
8
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1answer
123 views

Structure theorem for finitely generated commutative $semi$groups.

$\newcommand{\ZZ}{\mathbb{Z}}$ It is a classical theorem that a finitely generated commutative group is isomorphic to one of the form: $$ \ZZ^n \oplus \ZZ/{m_1}\ZZ \oplus \ZZ/{m_2}\ZZ \oplus \dots ...
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1answer
225 views

Quasicommutative semigroups.

A Semigroup is called quasicommutative if for all elements $a,b$ there is some $r≥1$ such that $$ab=b^ra$$ We know that every commutative semigroup is also quasicommutative, so we can make lots ...
8
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2answers
149 views

Name for a function whose image has smaller cardinality than its domain

I asked this question in the comments of this question, whose title would have done just as well for mine. But I suppose it should be a separate question. Is there a name for functions ...
8
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1answer
323 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 ...
7
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3answers
571 views

Is a semigroup $G$ with left identity and right inverses a group?

Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group? Now, If both the identity and the ...
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4answers
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Group identities and inverses

A set is a set. A magma is a set with a binary operator. A semigroup is a magma with an associative binary operator. A monoid has a two-sided identity. And a group has two-sided inverses. I am ...
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Is there a name for a semigroup whose idempotents form a subsemigroup?

For a semigroup $S,$ I will denote by $E(S)$ the set of all idempotents of $S$. For $X\subseteq S,$ let $X^2$ mean $\{xy\,|\,x,y\in X\}.$ Is there a name for the class of semigroups $S$ such that ...
7
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6answers
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What are the subsemigroups of $(\mathbb N,+)?$

While trying to solve a somewhat bigger problem, I realized that I don't know what the subsemigroups of one of the most important semigroups, $(\mathbb N,+)$, are. (I assume $0\not\in\mathbb N$.) I've ...
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1answer
252 views

When a semigroup can be embedded into a group

Under what assumptions can a semigroup $(S,*)$ be embedded into a group?
7
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296 views

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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2answers
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Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
7
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1answer
279 views

Weak generator of Feller semigroup

Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in ...
6
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5answers
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Is there an idempotent element in a finite semigroup?

Let $(G,.)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
6
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1answer
657 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 ...
6
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2answers
265 views

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?
6
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About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
6
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1answer
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About automorphisms of commutative semigroups

Suppose that $M$ is a commutative monoid and that the product $P$ of $M$ and the nonnegative integers $\mathbb{N}$ with addition has no nontrivial automorphisms. The set $S$ of pairs $(m,n)$ in $P$ ...
6
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1answer
231 views

Is the additive semigroup of natural numbers the multiplicative semigroup of a ring?

$\mathbb N$ will denote the set $\{0,1,2,\ldots\}.$ The semigroup $(\mathbb N,+)$ doesn't have a zero element. $\mathbb N^0$ will denote the semigroup $\mathbb N$ with zero adjoined, that is the set ...
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What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the ...
6
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0answers
129 views

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 ...
5
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2answers
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Does there exist a semigroup such that $(xy)^n = x^n y^n$ that is non-Abelian? If so, can this property be finitely axiomatized?

Suppose $S$ is a semigroup such that for all $x,y \in S$ and all natural $n$ we have $$(xy)^n = x^n y^n.$$ If $S$ group, then it is Abelian; indeed a stronger statement holds, see here. Does there ...
5
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3answers
117 views

Proving a binary operation on $\mathbb{Z}$ gives a semigroup

Let $x\circ y= x +y-xy, \quad (x,y) \in \mathbb{Z}$ where $\circ$ is a binary operation on $\mathbb{Z}$, prove that this is a semigroup. My Work To prove we have to check two things: $\mathbb{Z}$ ...
5
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1answer
488 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 ...
5
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1answer
168 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.
5
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1answer
190 views

Help me prove equivalently of regular semigroup and group.

Let $S$ be a semigroup. Prove that the following are equivalent: $\forall a \in S \exists! x \in S$ such that $ax \in E(S)$ where $E(S)$ is the set of all idempotent. $\forall a \in S \exists! x \in ...
5
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1answer
145 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
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1answer
68 views

A statement about an element $a$ in semigroup S, such that $aS$ containts idempotent and $a=axa$ implies $x=xax$

I have been currently studying some characteristics of completely regular and completely simple semigroups and I have came across a lemma, which seems simple, but I'm struggling with it's proof, so I ...
5
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1answer
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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 ...
5
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0answers
249 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 ...
4
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2answers
344 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 ...
4
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2answers
71 views

How to move from a right semigroup action to a left semigroup action?

Let $S$ be a semigroup and $X$ any set. Define a left action of $S$ on $X$ to be a map $\sigma: S \times X \rightarrow X$ with the property that $(st)x = s(tx)$, where we define $gx = \sigma(g,x)$ ...
4
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3answers
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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 $F$ ...
4
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1answer
167 views

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 ...
4
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1answer
47 views

One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
4
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1answer
72 views

Is there a standard name for this semigroup?

Given a semigroup $X,$ we can form a new semigroup $Y$ by asserting that: the carrier of $Y$ is the set $X^2$, and the law of composition in $Y$ is given by $(a,b)(a',b')=(aa',b'b).$ Finally, ...