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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If $G,H$ are monoids and not groups, prove that $f(e_G)=e_H$ may be wrong.

Problem: If $f:G\longrightarrow H$ is a homomorphism of groups, then $f(e_G)=e_H$ and $f(a^{-1})=f(a)^{-1}, \forall a\in G$. Show by example that the first conclusion may be false if $G, H$ are ...
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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 ...
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How many associative ternary operations there are on a finite set?

We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$ If $X$ is a set and and cardinality is $|X|=n$ then ...
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Example of an inverse semigroup

An inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...
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Evolution Equation

Let $X=L^2(0,\pi)$. Define the operator $(A,D(A)$ by: $$D(A)=\{u\in H^2(0,\pi):u(0)=u'(\pi)=0\} ,\quad \quad Au=u''$$ Show that $A$ is the infinitesimal generator of a $C_0$ semigroup of contractions ...
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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 ...
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Commutative Cancellative Semigroup: When is an irreducible element prime?

Suppose $(S,\cdot)$ is a semigroup with neutral element $e$ (i.e. $xe=ex=x$ for all $x\in S$) and the following properties: S is commutative: $xy=yx$. S is cancellative: $xy = xz$ implies $y = z$. ...
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Involution on semigroups with identity

I'm trying to understand the following: Let $S$ be a semigroup. By an involution on $S$ we mean a map $* : S \to S$ satisfying for all $a,b\in S$ $(ab)^*=b^*a^*$ $(a^*)^*=a $ My problem is the ...
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Does a discrete semilattice of finite width have finite number of points between any pair of elements.

Let $(S,<)$ be a countably infinite semilattice such that: 1) $S$ is no-where dense - i.e. there does not exist a subset $T$ such that for all $a,b\in T$ with $a<b$, there exists $c\in T$ with ...
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Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
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A construction using a semiring

In page number 3 of J.S. Golan's book Semirings and their applications, there is a result which says that if $R$ is a hemiring and $S$ is a subhemiring of $R$ which is a semiring having ...
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Can I define a category as a monoid with partially defined multiplication?

A groupoid can either be thought of as a category whose morphisms are isomorphisms, or as a generalization of a group whose multiplication is only partially defined. Can I do a similar thing with ...
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Semigroups — Underlying Set Non Empty or it can be Empty?

Many authors consider Semigroups being Non Empty sets. Others include empty sets as Semigroups. What is the rationale behind both choices ?
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under what conditions a product of matrices is the identity matrix (more complicated than that)?

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple ...
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semidirect product of semigroups

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function ...
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Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
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Commuting probability

http://www-rohan.sdsu.edu/~vadim/commute.pdf Where does the commuting probability formula on page 8 come from ?
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operator semigroups with negative growth bound.

Let $(T(t))_{t\geq 0}$ be a strongly continuous semigroup on a Banach space $X$, for which we assume that its growth bound $\omega_0$ is negative. Let $(A,D(A))$ be the generator of $(T(t))_{t\geq ...
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Understanding the definition of a Integral

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on ...
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Is Writing a Semi Group?

While writing this question, I compose letters from a set $L=\{a,...z,A,...Z\}\cup\{\;\text{ } \;\}$. Writing has a binary operation which is associative. The result always is an element of $L^n$. ...
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Proof of no identity element in a multiplication semi group

I have a question from my workbook: Let $\mathrm{E}$ denote the set of even integers. This forms a semi-group under multiplication. Show that there is no identity in this semi-group. Now this is ...
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An intersection equation in semigroups

Do you have an example of a semigroup $S$ and a collection of its subsets $(A_i)_{i\in I}$ and $a\in S$ such that $$a\big(\bigcap_{i\in I}A_i\big)\ne\bigcap_{i\in I}aA_i$$ ?
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Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
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Semigroups and units

I am working my way though Basic Algebra 1. I am currently on chapter 1 , and more specifically I busy with the following exercise: "Let $S$ be a set a define a product in $S$ by $ab=b $. Show that ...
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How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
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Prove that a cyclic semigroup is either finite or isomorphic to $\langle \mathbb{N},+ \rangle$

A semigroup $G$ is cyclic if $G$ is generated by a single element. I know that a finite cyclic group generated by $a$ is necessarily abelian, and can be written as $\{1, a, a^2, . . . , a^n−1\}$ ...
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Semilattices are congruence-semi-distributive

A semilattice $(S,\cdot)$ is a commutative idempotent semigroup. A congruence on a semilattice is an equivalence relation that preserves multiplication, i.e. $x_1\mathrel{\theta} y_1$ and ...
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Involution on inverse semigroups

I'm trying to prove the following for inverse semigroups $\bf Def:$ an inverse semigroup $S$is a semigruop such that for each $x\in S$ the exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. An ...
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Isomorphism of direct product of semigroups

I would appreciate some help with the following problem. Consider four semigroups $A,B,C,D$. I was able to prove that $A\cong C\wedge B\cong D$ implies $A\times B\cong C\times D$. But does also ...
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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 ...
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positive integer polynomial under the usual polynomial multiplication

consider the set of polynomials with positive integer coefficients together with the operation, usual multiplication of polynomials. now my first question is does this set together with the ...
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Proof of commutativity for topological I-semigroups

A topological semigroup on a closed interval I and order topology is called I-semigroup if 1 acts as an identity and 0 as an annihilator. I've seen in several articles that such I-semigroups should ...
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Findout if $\min(x,y)$ and $\max(x,y)$ are semigroup, monoid or group

I need to find out whether $A \oplus B := \max(A,B)$ or $A \ominus B := \min(A,B)$ form a semigroup, monoid or group for $\oplus : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and $\ominus : ...
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In a dagger category, what do we call an arrow $f$ such that $f \circ f^\dagger \circ f = f$?

In a dagger category, what do we call an arrow $f$ satisfying $f \circ f^\dagger \circ f = f$? In $\mathrm{Rel}$ (and, more generally, an allegory) this is straightforwardly equivalent to $f \circ ...
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Generalising cover maps from monoids to semigroups

Let $T,S$ be monoids. A partial surjective mapping $\psi : T \to S$ is called a cover map if for each $s \in S$ there exists some $\hat{s} \in T$ called a cover of $s$ such that for each $t \in ...
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How to calculate when a specific semigroup is closed under conjugation by this group.

Let $\mathcal{T}_n$ be the set of $n \times n$ matrices such that $X\in \mathcal{T}_n$ can be written as: $$X = \bigl( \begin{array}{cc} 1 & \overline{0} \\ T & \underline{0} \end{array} ...
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What does $\mathcal{J}$ stand for in Green relations?

Following this book The Algebraic Theory of Semigroups, Volume I , we see that: $a\mathcal{L}b$ means $a$ and $b$ generate the same principle left ideal of the semigroup $S$. $a\mathcal{R}b$ means ...
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Testing for the cancellation laws

When I'm given a presentation of a finitely generated semigroup/monoid, are there any tricks I could use to check if it is cancellative on both sides? I'm not asking for a general algorithm, as I ...
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Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
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looking for books on topological semigroup:

I'm looking for several books on topological semigroup: Topological semigroups: history, theory, applications. Karl Heinrich Hofmann Mathematics Research Library, Tulane University. The ...
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Naturally Ordered Semigroup: Why does axioms imply order of group is infinite countable ? Why are every group equal up to isomorphism?

Naturally Ordered Semigroup: http://www.proofwiki.org/wiki/Definition:Naturally_Ordered_Semigroup I know $\mathbb N$ is a naturally ordered semigroup by definition. Also I know that the naturally ...
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Partial differential equations and semigroups: explanation of an example.

The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem $$\left\{\begin{align*} ...
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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 ...
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Does Cartesian Product and Collection of all Sets Perform a Semigroup?

We know that the Cartesian Product is a binary operation. Also it is an associative operation. We know that Cartesian Product of two set is again set, there is even closure axiom. So I need to know ...
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If a semigroup satisfies these identities, is it necessarily commutative?

Suppose a semigroup $X$ satisfies the following identities. $$xya\equiv yxa,\quad axy \equiv ayx$$ Without assuming anything further, can we deduce that $X$ satisfies $xy \equiv yx$? In particular, ...
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Solution $\chi$ of $\chi_k(x)=\chi_x(f_k(x))$ given $f_k$ bijective

Given a family $\{ f_a \}_{a\in H}$ of bijective functions over a set $H$ I need to find a family of functions $\{ \chi_a \}_{a\in H}$ from $H$ to $H$ such that for every fixed $k$ in the family ...
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(Transformation) Semigroups, the semigroup $\mathbf D_n$ and the wreath product

I have some trouble understanding the following proof, were I can't even figure out how some terms are defined. But first I state some definitions and preliminary lemmas. A transformation semigroup ...
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Do cosets of this semigroup partition this ring?

Let $R = \Bbb{Z}^3$ be the usual direct product ring, and let $S = \{ (k^a, k^b, k^c) : k \in \Bbb{Z} - \{0\}\}$ be a subsemigroup of $R$. Then do the cosets $aS$ partition $R$? Let $R^* = S^{-1}R$ ...
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Unique idempotents in semigroups

In a finite semigroups every element has a unique idempotent power, just take $s^{n!}$ where $n = |S|, s \in S$. In an infinite semigroup there are clearly elements without idempotents, just take ...
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Are there interesting examples of medial non-commutative semigroups?

There exist semigroups $S$ (written additively) such that $S$ is medial, meaning $(a+b)+(a'+b') = (a+a')+(b+b')$. $S$ is not commutative. Example. The left (and right) zero semigroups are all ...