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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Multiplicative closure of convex set of matrices

Given a convex polytope of matrices $C$ is there a way to find its multiplicative closure $C^{*}$ such that $C^{*}$ is the smallest set which contains all products of sequences of matrices from $C$. ...
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Left Reversible Semigroups

A semigroup $S$ is left reversible if $aS \cap bS \not= \emptyset$ for every $a$, $b$ in $S$. Let $S$ be a nonempty left reversible semigroup and $S_0 \subseteq S$ be a proper subsemigroup of $S$. ...
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What are some applications of transformation semigroups?

I have been told that transformation semigroups have applications to statistics, computer science, and combinatorics. What are some basic (and if possible, simple) examples of transformation ...
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Subsemigroup of permutable semigroup!

Let $S$ be a semigroup and $n$ integer greater than 1. A sequence $s_{1}, s_{2},...,s_{n}$ of $n$ elements of $S$ is called permutable if the product $s_{1}s_{2}...s_{n}$, remains invariant, under ...
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Nonminimal idempotents

Let $S$ be a semigroup with operation $+$. Note that we do not assume $S$ is abelian. We say that $I\subset S$ is a left ideal if and only if for all $s\in S$, $s+I\subseteq I$, and we say that an ...
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What does it mean when people say that groups are a study of symmetry?

I see many people make remarks to the effect that groups have basic symmetry properties. I am familiar with Cayley's Theorem and the symmetric groups $S_N$. However, when I think of the symmetric ...
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1answer
51 views

Finitely generated Clifford semigroup

If $S$ is a finitely generated Clifford semigroup and $S'$ a subsemigroups of $S$, how can I prove that $S'$ is also finitely generated?
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In a commutative semigroup, $a_{\phi(1)}a_{\phi(2)}…a_{\phi(n)}=a_{1}a_{2}…a_{n}$

In the book of Clifford and Preston,"The algebraic theory of semigroups" volume I, I am reading this: If $a_{1},a_{2},...,a_{n}$ are elements of a commutative semigroup $S$ and $\phi$ is any ...
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Isomorphism in Clifford semigroups

If I take a subsemilattice $S'$ of the Clifford semigroup $S$, such that the groups of $S'$ are abelian by finite of finite index in the respective groups of $S$ and I want to prove that $S$ is ...
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S an inverse semigroup with semilattice of idempotents E and $\sigma$ the minimum group congruence on S.

Let S be an inverse semigroup with semilattice of idempotents E, and let $\sigma$ be the minimum group congruence on S. Show that the following statements are equivalent: (a) $x\sigma y$; (b) $(\...
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D-􀀀classes in an inverse semigroup are ‘square’.

Show that the $\mathcal{D}$-classes in an inverse semigroup are ‘square’. More precisely, show that there is a bijection from the set of $\mathcal{L}$-classes in a $\mathcal{D}$-class $D$ onto the set ...
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Is a subsemigroup of a Clifford semigroup also a Clifford semigroup?

Is a subsemigroup of a Clifford semigroup also a Clifford semigroup? I can prove that the maximal group image of a Clifford semigroup is a Clifford semigroup, but I am not sure whether any ...
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51 views

Semigroup of matrices and expander Cayley graphs

I am interested in proving or disproving that certain Cayley graphs are expander. Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & ...
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1answer
87 views

Finding ideals of a finite semigroup machinery (GAP)

As you see, I did a finite semigroup and then try to find its possible ideals in a very basic way (with the hand of a friend): ...
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18 views

scalar valued semigroup vs vector valued semigroup

I am reading a chapter on heat equation, semigroup in the PDE book written by Jost. Some confusions arises when a 1D problem generalised to multidimensional. In the bottom right of the following, i.e. ...
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1answer
33 views

Basic inequality related to semigroup property

Let $T$ be a one parameter semigroup on Banach space $X$. We know that $T$ has the property $$T(t+s)=T(t)T(s)\quad\text{for all}\quad t,s\ge 0. $$ I was reading some notes on evolution equations and ...
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Can a non-finitely generated semigroup be a subsemigroup of a finitely generated one?

Sorry! The answer doesn't seem obvious at all to me... If $\langle S \rangle \leqslant \langle T \rangle$ can $S$ be infinite while $T$ finite? I think the answer is yes. Let $\Bbb{Z}^{\times} \...
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$\langle ab + 1 : a,b \text{ prime}\rangle$ is not a finitely generated subsemigroup of $\Bbb{Z}^{\times}$.

Let $T \equiv PP + 1 \equiv \{ ab + 1 : a,b \text{ are prime }\} \subset \Bbb{Z}^{\times}$. Consider the subsemigroup generated by $T$. How can I show that it is not finitely generated, by that I ...
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$\#((P_n + 1) \cap P_l) = \infty$ where $P_n = \{ z \in \Bbb{Z}: $ that involve only the first $n$ primes $\}$.

Let $P_n = \{ \pm p_1^{e_1} \cdots p_n^{e_n} : e_i \geq 0, p_j = j$th prime $\}$. Define $P_0 = \{\pm 1\}$. Then: $\Bbb{Z}\setminus \{0\} = \bigcup_{n\geq 0} P_n$ $P_n \subset P_{n+1}$ $P_n \cdot ...
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Infinite free commutative semigroup

Definition from P.A.Grillet, "Abstract Algebra" (quoted with modifications): The free commutative monoid on a finite set $X=\{x_1, x_2,..., x_n \}$ is the semigroup of all monomials $x^{a_1}_1 x^{...
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How do the equal equivalence classes under the relation defined below imply the following?

First, we have a commutative semigroup $(S,*)$. Now let $(S_1,*_{↾_1})$ and $(S_2,*_{↾_2})$ be its two subsemigroups, with $*_{↾_1}$ and $*_{↾_2}$ being the restrictions of $*$ to $S_1$ and $S_2$. ...
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How does $0$ being an identity element of naturally ordered semigroup follow from its definition?

In the context of naturally ordered semigroups, $0$ is defined as the semigroup's smallest element (in other words, ${\forall}x\,{\in}\,S:0\,{\preceq}\,x$ where $ S$ is our semigroup). Natural ...
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How do the axioms of naturally ordered semigroup imply commutativity?

Semigroup $(S,*)$ is naturally ordered (natural ordering is supposed to behave like natural numbers under addition) iff: $1$. ${\forall}I\,{\subseteq}\,S:{\exists}m\,{\in}\,I:{\forall}n\,{\in}\,I:m\,{...
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1answer
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Width of Join Semilattice in $\mathbb R^n$

Suppose $S$ is a join semilattice that can be generated by $m$ generators. I would like to know an asymptotic upper bound on the width of $S$ (the maximum cardinality of all antichains in $S$) under ...
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1answer
42 views

Does this semigroup have an identity element?

Let G be the set of functions that map {1,2,3,4} into {1,2}, the binary operation is the usual composition of mappings and G is a semigroup. From my knowledge, I would say that it doesn't have an ...
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1answer
34 views

Let $S$ be a regular semigroup, $\phi:S \rightarrow T$ an epimorphism… Prove that $\phi(a)=c$

Let $S$ be a regular semigroup, $\phi: S \rightarrow T$ onto morphism of semigroups, $c,d \in T$ mutually inverses, ie, $c=cdc$ and $d=dcd$. Suppose that $c=\phi(x)$ and $d=\phi(y)$, where $x,y \in S$...
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Does subsets of a semigroup with strictly smaller cardinality insure infinitely many disjoint translation copies of the set?

Let $S$ be a semigroup with infinite cardinality, $A\subset S$ with $|A|<|S|$. Under what condition we may find a infinite net $\{s_\alpha; \alpha\in \Gamma\}$, such that $s_\alpha A \cap s_\beta A ...
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1answer
72 views

Are some results about coloring positive integers valid for other semigroups?

There are some results in Ramsey theory, which involve additive structure of $(\mathbb N,+)$. For example, if we color the set $\mathbb N$ by finitely many colors, then: There are three numbers $x$,...
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Subsemigroup of integers is finite type?

I think every subsemigroup of semi-group $(\mathbb N,+)$ is generated by finite many integers (this is not true for $\mathbb N^2$). Remind: subsemigroup $=$ subset stable for $+$. Can you give short ...
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Why isn't $\overline{\pi}$ a bijection between sets of cardinality $159$ and $\aleph_0$ respectively? [duplicate]

Earlier today, I (essentially) asked the following: (Faulty) Question. Given a set $X$, write $F_\mathbf{Semi}X$ for the semigroup freely generated by $X$. Suppose $X$ is a finite set, $n$ ...
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Seeking more information regarding the function $\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right].$

Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows. $$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$ The motivation is that according to ...
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Membership in Semigroup Generated by a Set of Matrices

I have a mathematical problem which can be expressed in the following form. I am given a small number (less than 10) of typically 50x50 matrices. Each column of each of these matrices has one 1 and ...
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1answer
26 views

Let $S$ be a semigroup that satisfies the property $\forall a \in S, aS=S \wedge Sa=S$. Show that $S$ is a group.

Let $S$ be a semigroup that satisfies the property \begin{align*} \forall a \in S, \quad aS=S \wedge Sa=S. \end{align*} I want to show that $S$ is a group, ie, that $S$ satisfies (1) $\...
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103 views

Viewing Semigroups as Categories?

I am wondering how to view semigroups as categories. For example, we can easily view monoids as categories with a single object. Unfortunately, semigroups do not necessarily have identities, so the ...
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Is a submonoid of a commutative, finitely generated monoid, always finitely generated? [closed]

Let $M$ be a commutative, finitely generated monoid and $N$ its submonoid. Is $N$ finitely generated as well?
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Examples of left-topological compact semigroups

I was reading chapter in Todorčević's book Topics in Topology (LNM 1652, DOI: 10.1007/BFb0096295) which deals with the semigroup $\beta\mathbb N$. Several results about left-topological compact ...
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Is every semigroup with (possibly non-unique) division a group?

Let's say that a semigroup $(S,\cdot)$ has weak division if for all $a,b\in S$ there exist $c,d\in S$ such that $ac = b$ and $da = b$. Note that we don't require $c$ and $d$ to be unique. This ...
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2answers
87 views

Equivalent definition of abelian group

Assuming that given operation is commutative and associative, are following definitions equivalent? 1) A zero element $e_0$ exists such that $e_i+e_0=e_i$ and inverse $e_{i'}$ exists for every $e_i$ ...
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1answer
31 views

Show that $\lim \frac{f_\epsilon -1}{\epsilon}$ is purely imaginary if each $|f_\epsilon| = 1$

For $0< \epsilon < 1$, suppose we have complex numbers $f_{\epsilon}$ such that each $|f_{\epsilon}| =1$ and $$ \lim_{\epsilon \to 0} \frac{f_\epsilon - 1}{\epsilon} := a $$ exists. Prove ...
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what is a symmetric semigroup?

First, I think I know what is a symmetric group roughly from algebra. The group of permutation on a set with $n$ element is denoted by $S_n$, and called the symmetric group on $n$ elements (or $n$ ...
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Nowhere commutative semigroups = rectangular bands?

I have recently read about Nowhere commutative semigroups, and there, they say, if $S$ is a semigroup, then these statements are equivalent: $S$ is nowhere commutative($ab=ba$ implies $a=b$). $S$ is ...
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How to prove that this sentence is true? [duplicate]

I have the following task: Prove that a semigroup $S$ is a rectangular band if and only if it is nowhere commutative. So, if I get it right, what I have to prove has $2$ sides. First, I need to ...
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1answer
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Proving an identity of the resolvent set

$\{T(t)\}_{t\ge 0}$ is a $C_{0}$-semigroup with infinitesimal operator $A$. I'm trying to prove that the set $\{ z|\text{ Re }z>\omega_{0}\}$ belongs to $\rho(A)$, and for $z$ in this set, the ...
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1answer
29 views

Foundation Semigroups Examples

I recently started looking into the subject of foundation semi-groups and I'm trying to find simple (none-trivial) examples of foundations semi-groups. I know that every locally compact group is ...
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1answer
71 views

A simple $C_{0}$-semigroup question.

Let $u:[0,t_{e}]\to\mathcal{D}(A)$ satisfy $$\begin{cases} \frac{du}{dt}=Au & 0\le t \le t_{e} \\ u(0)=x \end{cases}$$ I want to prove that necessarily $u(t)=T(t)x$. So it's clear to see that $...
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Equivalent Norms for Intermediate Subspaces

Let $(X,\left\|\cdot\right\|)$ be a Banach space, and let $\left\{T(t) : t\geq 0\right\}$ be an equibounded strongly continuous semi-group on $X$. Define a functional $\left\|\cdot\right\|_{\alpha,r;q}...
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Is my proof of the following correct?

Question: Let $M,P,Q$ be semigroups and $\sigma:M\rightarrow P$, $\rho:M \rightarrow Q$ be morphisms with $\sigma$ surjective. Then $\ker \sigma \subseteq \ker \rho$ if and only if $\rho$ factors ...
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spectrum of weighted translation semigroup

My Banach space is $\mathcal X=\rm{L}^1(\mathbb R_+)$. I would like to know the spectrum of $A\phi(x)=-\phi'(x)-f(x)\phi(x)$ on $D(A) = \{g\in\mathcal X,\ g\text{ absolutely continuous}, g(0)=0\text{ ...
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1answer
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Free idempotent semigroup with 3 generators

On Finite free objects I asked examples of finite free objects. I got answer "The free idempotent semigroup satisfies $x^2=x$ for each element $x$. And I confirm it is finite if it is finitely ...
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1answer
64 views

Computing the shortest encoding for a transformation

Here goes... Let $n = 2^p$ for some $p \in \mathbb{N}$. Let $\mathcal{T}_n$ denote the set of all transformations on $\mathbb{Z}_n$ (viz. the transformation monoid). Pick $C$ to be an order $|\...