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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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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35 views

Showing an operator is linear

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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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 ...
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1answer
23 views

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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567 views

Semigroups, monoids, & groups!

I need help determining if these are semigroups, monoids, or groups? a) $\mathbb Z ^+$, where $\#$ is defined as ordinary multiplication b) $\mathbb Z ^+$, where $a \# b$ is defined as $\gcd(a,b)$ ...
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561 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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187 views

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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47 views

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 ...
6
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126 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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240 views

Inverse of the Grothendieck group construction?

Is there an "inverse" of the Grothendieck group construction that would generate a (somehow "simplest") Abelian semigroup given some (suitably qualified, if necessary) Abelian group? (I realize that ...
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63 views

Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
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89 views

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 ...
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20 views

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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95 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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124 views

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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129 views

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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56 views

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 ...
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72 views

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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21 views

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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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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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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fraction power of operators in $C_0$ semigroup.

Let $X$ be Banach space, and $\{Z(t)\}_{t\geq 0}\subseteq B(X)$ be the $C_0$-Semigroup of operators defined on $X$. Moreover, let $A$ be the infinitesimal generator of $\{Z(t)\}_{t\geq 0}$. A ...
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Ergodicity and the generator.

For a markov process $X_t$ adapted to ${\cal F}_t$, let $A$ satisfy $$\lim_{h\longrightarrow 0}[E\{f(X_{t+h})|{\cal F}_t\}-f(X_t)]/h = Af(X_t).$$ What are (references to?) conditions that allow one ...
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27 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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47 views

What operations is this set closed under? The set of all $\{(a,b) : (a,b) \in \Bbb{Z}^2, \gcd(a,b) \neq 1\}$.

Let $S = \{(a,b): a,b \in \Bbb{Z}, \gcd(a,b) \neq 1 \}$. Under what binary operations is $S$ closed (that we can come up with)? I came up with the following: $$ \begin{align*} (a,b)(c,d) = \\ ...
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45 views

A combinatoric theorem about semigroups

I am struggling with the following theorem about semigroups, so I was hoping someone could give me a hand. The theorem states: "Let $S$ be an arbitrary semigroup such that for every $a\in{}S$ it ...
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56 views

Strong continuity in time vs uniform continuity in time

I have a problem with understanding the definition of strong continuity and uniform continuity for the families of operators, e.g. semigroups. Let $(X_t)_{t \geq 0}$ be a family of bounded linear ...
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63 views

irreducible words in a semigroup

Let $S$ be a monoid generated by $\{x_1,x_2,x_3,x_4,x_5\}$ with the following relations: $x_i^2=0$ for all $i$, $x_ix_j=x_jx_i$ for all $|i-j| \geq 2$ and $x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}$ for all ...
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47 views

Trivial question about semigroups

I have a question about analytic semigroups. I know that if $A: D(A)\subset X\to X$ is a sectorial operator on $X$, it generates an analytic semigroup $e^{tA}$. One of the properties of this semigroup ...
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25 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
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138 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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143 views

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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universal semigroup via relations

My question is about a paper that constructed a semigroup from a group $G$ as follows: let $[g]$ are choosen from a set having as many elements as $G$ and let $S(G)$ be the universal semigroup via the ...
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What are $\Gamma$-semigroups?

I have some problems with $\Gamma$-Semigroups, the definition that I've found is A $\Gamma$-Semigroup is a pair $(M,\Gamma)$ defined as follow If $x,y$ and $z$ are in $M$ and $\alpha$ and ...
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What is the semigroup reduct of an abelian torsion group

I was going through a paper on universal algebra where the author mentions one of his examples as a the "semi group reduct of an abelian torsion group". I have no idea what a semigroup reduct means ...
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partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
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36 views

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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59 views

About the multiplicity of a semigroup ring.

Let $A=K[X^{n_1}, \dots, X^{n_s}]$, $S=\langle n_1, \dots, n_s \rangle$ ($n_1 < \cdots < n_s$) a numerical semigroup and let $m$ be the maximal ideal $(X^{n_1}, \dots, X^{n_s})$. We have to ...
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57 views

Principal ideal in a semigroup ring.

Let $S= \langle(0,1),(3,2),(5,2) \rangle \subset \mathbb{N}^2 $ be a semigroup and consider the semigroup ring $K[S]$, with $K$ a field. We can consider the principal ideal generated by ...
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Proof involving handling of Greens Relations (Semigroup-Theory)

I am working through a proof which involves Green's relations. It goes like this (for $s,s', e', e$ holds $se = s, e^2 = e$ and $s'e' = s', e'^2 = e'$, they are called linked pairs) Conversely, ...
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How to find the quotient group given a specific equivalence relation on $\mathbb{N}^3$

Let $L=\mathbb{Z}\{(3,4,-5)\}$ and let $$ \displaystyle Q \cong \mathbb{N}^3/\sim_L $$ where $\sim_L$ is the equivalence relation on $\mathbb{N}^3$ given by $\mathbf{u} \sim_L \mathbf{v} \iff ...
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25 views

Proper term for a monotonic magma

Let $M$ be the multiplication table for a finite magma. Entries are labeled $0,\ldots,n-1$. M has the property that $M(i,j)\ge i, M(i,j) \ge j$. What is the proper term for this kind of magma and ...
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59 views

Generalization of $e^{t A}$ to $e^{t^{\alpha}A}$

I would like to give a mathematical structure of an operator of the form $(S_{\alpha}(t))_{t \geq 0}$ where $S_{\alpha}(t)=e^{t^{\alpha} A}$ as it was done for the operator $S(t)=e^{t A}$ for ...
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75 views

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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79 views

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 ...