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. This tag is most frequently used for questions related to the concept of semigroups in the context of abstract algebra. Please use the more ...

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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 $gx = \sigma(g,x)$ ...
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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 ...
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3answers
118 views

How are the powers being changed

I have a semigroup $S$ including a generator, say $d$, such that $$d^4=d$$ I am trying to guess the general rule of $d$'s powers such that when I want to calculate $d^n, n\in\mathbb N$; I can simplify ...
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4answers
339 views

Inverse elements in the absence of identities/associativity.

Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows. $(xy)z=x(yz).$ $xe=ex=x.$ ...
3
votes
1answer
182 views

Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
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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: $\mathbb{Z}$ ...
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2answers
161 views

In a semigroup, $ab=ba\Rightarrow (ab)^k=a^kb^k$.

Let $a,b \in S$ where $(S, *)$ is a semigroup and suppose $ab = ba$. Prove that $(ab)^k = a^kb^k$ for all $k ≥ 1$
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vote
1answer
25 views

inverse on topological semigroup

Assume $(G,\cdot)$ denotes a topological semigroup (no id, non-commutative). Let $V\subset G$ be open and take some arbitrary $g\in G$. Define $$gV^{-1}:= \bigcup_{x\in V}gx^{-1} = \bigcup_{x\in ...
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Unitary operators - convergence problem

Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$. Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
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2answers
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$\mathcal{D}$-classes

Let $$\alpha = \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}} 1&2&3\\ 1&3&3 \end{array}\right) \in \mathcal{T}_3\text{.}$$ (a) Show that the ...
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Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
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2answers
168 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 ...
3
votes
1answer
278 views

Rees matrix semigroup

Let $\mathcal{M}^0 = \mathcal{M}^0 ( G; I , \Lambda ;P)$ be a Rees matrix semigroup ($G$ a group, $I$, $\Lambda$ non-empty sets, $P=(p_{\lambda i})$ a $\Lambda \times I$ matrix over $G \cup \{0\}$ ...
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1answer
70 views

A regular quasicommutative semigroup $S$

Here is a theorem: I could go inside the theorem and know some few points of it. It's told that If all elements of $H_e$ are of finite order so the group $H_e$ is Hamiltonian. My question ...
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1answer
100 views

Semigroup homomorphism and the relation $\mathcal{R}$

Let $S$ be a semigroup and for $a\in S$ let $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The relation $\mathcal{R}$ on a semigroup $S$ is defined by the rule: $$a\;\mathcal{R}\; ...
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1answer
167 views

Semigroups and relations

Recall that if $S$ is a semigroup then for $a\in S$ $$Sa = \{sa : s \in S \}\text{,}\;\;\;S^1 a = Sa \cup \{a\}\text{,}$$ $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The ...
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1answer
150 views

Rings and idempotent semirings

If $\mathbb{R}$ is the real numbers and x#y=max{x,y} then ($\mathbb{R}$,#,+) is a semiring where ($\mathbb{R}$,#) is a semigroup and + distributes over #. If you have a set R with three distinct ...
8
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1answer
246 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 ...
9
votes
1answer
301 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 ...
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votes
1answer
369 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 ...
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votes
0answers
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A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
4
votes
1answer
292 views

Schrödinger Kernels on manifolds

Let $M$ be a compact Riemannian manifold and $\Delta$ be the Laplace-Beltrami operator. It is well-known that the solution operator to the heat equation $e^{t \Delta}$ is smoothing for $t>0$ and ...
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1answer
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Definition of tautological action

What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions? For reference the particular sentence is: "A ...
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1answer
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Semigroup congruence defined on generators

Let $S$ be an abelian semigroup generated by a nonempty set $X$. Suppose $\sim$ is an equivalence relation on $S$ satisfying the condition: "If $x \sim x'$ and $y \sim y'$ then $x+y \sim x'+y'$ for ...
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votes
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Is every sufficiently large semigroup $S$ a subsemigroup of the transformation semigroup on $S$?

In this question, all sets are finite. A semigroup is a set equipped with an associative binary operation, which I will call $\circ$. The tranformation semigroup on a set $X$ is the semigroup of maps ...
2
votes
1answer
86 views

Semigroup question

I am looking for the technical term for an element of a transformation semigroup that sends everything to one state. The best term I could think up was filter. For those that don't know a ...
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2answers
139 views

hint request: counting the elements in a subset of a numerical semigroup

I'm working on a problem for an online judge and I'm stuck. I'd like a nudge in the right direction (not an outright solution, please), relevant references, theorems, etc. After reading through ...
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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 $b*a=c*a$, ...
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Semigroup with exactly one left(right) identity?

Are there any examples of a semigroup (which is not a group) with exactly one left(right) identity (which is not the two-sided identity)? Are there any “real-world” examples of these (semigroups of ...
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1answer
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Is this a series convergence criterion? (Related to the Hilbert-Waring theorem)

By the set of natural numbers I will mean $\mathbb N=\{n\in\mathbb Z\,|\,n\geq0\}.$ I have come across a condition on a sequence $(a_n)_{n=0}^\infty$ of natural numbers that I feel may imply that ...
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1answer
189 views

Find identity element, invertible and inverses in $T=\mathbb Z \times \mathbb Q$

Let the following operation be defined on $T=\mathbb Z \times \mathbb Q$: $$\begin{aligned}(a,b)\centerdot (c,d) = (-ac, b+d+2) \end{aligned}$$ in the commutative semigroup $(T, \centerdot)$, find ...
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1answer
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Question on Malcev's _Immersion of an Algebraic ring into a skew field_.

I'm reading the paper Immersion of an algebraic ring into a skew field by Malcev. Doi: 10.1007/BF01571659, GDZ. On the third page of the paper, he writes that If $\alpha\beta\sim\gamma\delta$ and ...
3
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2answers
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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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votes
2answers
149 views

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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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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0answers
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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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0answers
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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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votes
3answers
157 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
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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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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
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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
174 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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2answers
222 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
260 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
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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
395 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
188 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 ...