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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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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16 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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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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2answers
28 views

$\langle ab + 1 : a,b \text{ prime}\rangle$ 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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14 views

$\#((P_n + 1) \cap P_l) = \infty$ 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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1answer
22 views

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 ...
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26 views

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 $S_2$. ...
2
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1answer
43 views

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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34 views

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: $1$. ...
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1answer
26 views

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
36 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
32 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 ...
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0answers
41 views

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 ...
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1answer
32 views

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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0answers
51 views

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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0answers
60 views

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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0answers
39 views

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
24 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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1answer
98 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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1answer
38 views

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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53 views

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 ...
3
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1answer
60 views

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
84 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 $e_i$ ...
2
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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 ...
2
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1answer
50 views

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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1answer
36 views

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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26 views

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
37 views

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
27 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 ...
3
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1answer
70 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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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 ...
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2answers
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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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22 views

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{ ...
2
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1answer
82 views

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 ...
2
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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 ...
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2answers
65 views

Are there any other useful semigroups aside from $e^{At}$

$e^{At}$ trivially satisfies both properties of semigroup namely $T(t+s) = T(t)T(s)$ $T(0) = I$ Does there exist any other commonly used operators aside from $e^{At}$ that is a semigroup? ...
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1answer
45 views

Build a Markov process from a transition semigroup

Suppose that we have a Markov process $X$ with time-homogeneus transition probability function given by $p(t,x,A)$, with $t>0, x\in E, A \in \mathcal E$, where $(E,\mathcal E)$ is the state space. ...
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Order of calculation in a Semigroup.

If $(A,\cdot)$ is a semigroup, i.e. if we have: $\forall (x,y,z)\in A^{3}, (x\cdot y)\cdot z=x\cdot(y\cdot z)$, then the order of calculations doesn't matter no matter the number of factors and we can ...
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493 views

Is there a name for this property?

Is there a general name for the following properties, (similar to the properties of existence of an additive identity, existence of multiplicative identity etc): For any given set, the intersection ...
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1answer
268 views

Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very ...
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Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
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1answer
155 views

Let $G$ be a semigroup. If for any $a,b\in G$, the equations $ax=b$ and $ya=b$ are solvable, then $G$ is a group.

I at a loss here. Showing that $G$ has an identity is the difficult part here. Obviously for each $a\in G$ there are $b,c\in G$ such that $ab=a$ and $ca=a$, but I need a hint as to how to prove these ...
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1answer
51 views

The identity elements of two multiplication satisfying interchange law.

Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 ...
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3answers
71 views

A semigroup with three or four elements without identity [duplicate]

Does there exist a semigroup with three or four (or finite) elements, without identity? I tried to construct such an example, but every example I tried to construct had an identity element.
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60 views

What is the right (a good) definition for a dual monoid?

Suppose we have the free abelian monoid $S = \{a^m : m \in \mathbb{N}_0\}$ on the set of one element $X = \{a\}$. The binary operation on the monoid is denoted by $\cdot$. If $(T,\ast)$ is another ...
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35 views

Ellis semigroup

I have only seen the construction of the Ellis semigroup applied to a group endowed with the discrete topology. Does the same idea works for any topological group? If not, where does it go wrong? (I ...
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1answer
48 views

Question about kinda-sorta identities in algebra

A friend showed me some semi-groups which were constructed as follows. Let $Q = \{ 1, \ldots, n \}$ be a finite set, and let the operation $\cdot$ be defined on $(Q^{2})^{2}$ by \begin{align*} ...
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26 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...