Tagged Questions

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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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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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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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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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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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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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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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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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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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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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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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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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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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 $... 2answers 79 views Equivalent Norms for Intermediate Subspaces Let$(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}...
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 ...