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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Tell if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it

Let the operation $\odot$ be defined in $\mathbb Z_6$ as follows: $$a \odot b = a +4b+2$$ check if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it. This is the way I ...
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What should I call this commutative monoid of order three?

I'm looking for a name for the monoid given by the following table: $$ \begin{array}{c|ccc}&1&a&b\\ \hline 1&1&a&b\\ a&a&1&b\\ b&b&b&b \end{array} $$ ...
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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 ...
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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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Is there a name for a subset $S$ of a group or a semigroup such that every two elements of $S$ commute?

Let $G$ be a group and $S$ its subset. I would like to consider the following condition on $S$. For every $x,y\in S,$ we have $xy=yx.$ This is trivially equivalent to $S\subseteq C(S).$ The ...
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Is the universal inverse semigroup of a commutative semigroup an embedding?

The question of existence of a universal inverse semigroup of an arbitrary semigroup has been answered before (this is a construction similar to the Grothendieck group). Let's refer to the universal ...
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155 views

What are some (or even one) interesting examples of (non-group) semigroups?

I'm going to give a lecture on Alon and Schieber's Tech Report on computing semigroup products (Optimal Preprocessing for Answering On-Line Product Queries). Basically, given a list of elements ...
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1answer
126 views

If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?

This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...
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2answers
352 views

Can we extend the definition of a homomorphism to binary relations?

This is going to be quite a long post. The actual questions will be at the end of it in section "Questions." INTRODUCTION After receiving an answer to this question about extending the definition of ...
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1answer
67 views

What condition on $m$ turns this semigroup into a monoid?

Suppose you have a monoid $(M,p,1)$ (viewing it as a triple of a set $M$, operation $p$, and unit $1$). Then for some $m\in M$ we can define a new product $p_m$ in $M$ by $p_m(a,b)=amb$. It's easy to ...
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1answer
172 views

Semigroups of matrices with zeroes and a single 1

I stumbled upon this while reviewing a Harvard lecture on abstract algebra. What I want to know is if these semigroups are known and, if so, what they are called. I've checked the assertions below for ...
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220 views

If every element of a monoid is right invertible, then every element is invertible

Let $G$ be a set with associative binary operation and a unit. Assume that for every $g\in G$ there exists $x \in G$ with $gx = 1$. Prove that $xg = 1$ is a consequence.
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1answer
247 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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2answers
2k views

If cancellation laws hold, then a finite semi-group is a group [duplicate]

Show that if both cancellation laws i.e $w.a = w.b \implies a = b$ and $a.w = b.w \implies a = b$ holds then a finite semi-group (a finite set with associative binary operation) is a group. I have ...
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1answer
375 views

Why isn't there much study on right groups (left groups)?

This question is related to and inspired by the question Why are groups more important than semigroups?. I am curious why I don't see much studies on right groups. On Pp.37 of Clifford& ...
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1answer
179 views

Errata for Semigroups and Combinatorial Applications by G. Lallement?

I am asking this question after years of frustration with the typos in the subject book I have read. It has been cited and referenced by many literatures and books in math and computer science. ...
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Constructing all semigroups over a given set without backtracking

Is there a procedure to construct all semigroups over a given set without backtracking? (Edit: see also how many associative binary operations are there on a finite set and ratio of semigroups over a ...
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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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313 views

give a counterexample of monoid

If $G$ 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
135 views

Two elementary question on automaton and language

1.What is the definition for a semigroup(or monoid) recognizing a set of words(or language)?2.Are recognizable,rational and regular equivalent to each other with respect to a language? PS:The reason ...
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110 views

How to ensure the syntactic semigroup of $X$ is the smallest semigroup recognizing $X$

Show that the syntactic semigroup of $X$ is the smallest semigroup recognizing $X$ in the sense that, for every semigroup $S$ recognizing $X$, there exists a morphism from $S$ onto the syntactic ...
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114 views

Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?

This question is a by-product of this one. I'm asking it because of this comment by Tara B. I'll repeat the definitions. The full transformation semigroup $\mathscr T_X$ on a set $X$ is the semigroup ...
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172 views

Idempotent separating congruence of an inverse semigroup.

Please can sombody help me with the proof of this lemma, or even a construction of the proof? I will be glad for that. Lemma: Show that if $\rho$ is an idempotent separating congruence of an ...
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372 views

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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2answers
238 views

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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214 views

$\mathcal{R}$-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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2answers
226 views

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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244 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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1answer
367 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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133 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 ...
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1answer
158 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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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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2answers
303 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?
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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
137 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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679 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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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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1answer
245 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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1answer
825 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 ...
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2answers
473 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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222 views

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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180 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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1answer
114 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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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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395 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
59 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 ...