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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some questions about ideals [on hold]

I am educated in the field of mathematical analysis. I need some information about semigroups: What are all ideals of $(R^+ , +)$. {I think that $(0,a)$} and also of $[0,1]$ with multiplication ...
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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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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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Congruence lattice of a semilattice is meet-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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The semigroup of powers of the differential operator in fractional calculus.

In my ignorance I'm slightly wary of a follow-up question here as it might belong in MO. If so, I'm sorry. Motivated simply by curiosity and this question, I'd like to investigate the semigroup $S$ ...
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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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26 views

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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32 views

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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1answer
33 views

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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1answer
52 views

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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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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1answer
45 views

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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1answer
60 views

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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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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42 views

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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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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49 views

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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43 views

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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39 views

(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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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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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 ...
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Is there a finite commutative semigroup $S$ with $S^2 = S$ which is not a monoid?

Is there a finite commutative semigroup $S$ with $S^2 = S$ which is not a monoid? where $S^2=\{ab\mid a,b\in S\}$. As is known, if such $S$ can be a ring with an addition then it is a monoid? So if ...
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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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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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Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...
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On the powerset of a monoid, does the second operation play nice with the first? Indeed, is it even useful?

Let $X$ denote a monoid. Then we can make $Y = \mathcal{P}(X)$ into a monoid, too. Define $$AB = \{ab \mid a \in A, b \in B\}$$ for all $A,B \in Y.$ We see immediately that $1$ (shorthand for $\{1\}$) ...
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27 views

If $S$ is a finitely generated periodic semigroup with the permutation property, then $S$ is finite.

In A. Nagy's Special Classes of Semirings, the first theorem is: Theorem 1.1 A finitely generated semigroup is finite iff it is periodic and has the permutation property. The definitions are as ...
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Does there exist a semigroup such that $(xy)^n = x^n y^n$ that is non-Abelian? If so, can this property be finitely axiomatized?

Suppose $S$ is a semigroup such that for all $x,y \in S$ and all natural $n$ we have $$(xy)^n = x^n y^n.$$ If $S$ group, then it is Abelian; indeed a stronger statement holds, see here. Does there ...
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Does there exist a semigroup such that every element factorizes in this way, which nonetheless lacks a left identity?

If a semigroup $S$ has a left identity-element, then for any $y \in S$ we can write $y = xy$ for some $x \in S$. Just take $x$ to be any of the left identities, of which there is at least one, by ...
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A semigroup $S$ that $|S| > 2$ is regular if and only if $x^2=x$ for any $x \in S$ where $S-\{x\}$ is a group

A semi-group $S$ is called regular if for any $y \in S$ there exists $a \in S$ such that $yay=y$. Let $S$ be a semi-group with more than two elements and $x \in S$ be such an element that $S - ...
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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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48 views

Necessary and sufficient conditions for the embeddability of a semigroup in a group

According to wikipedia, The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in (Malcev 1939).[5] Though theoretically important, the ...
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Is the set of all uniquely invertible elements of a semigroup an inverse subsemigroup?

For a semigroup $S$ and $x\in S$, an element $y\in S$ is called an inverse of $x$ iff $xyx=x$ and $yxy=y$. $S$ is called an inverse semigroup when every element of $S$ has a unique inverse. Every ...
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Does there exist a commutative magma such that $\mid$ is transitive, which is not a semigroup?

Let $M$ denote a commutative magma, and write $x \mid y$ iff $xa=y$ for some $a \in M$. If $M$ is a semigroup, then $\mid$ is transitive. Does there exist a commutative magma such that $\mid$ is ...
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If $S$ is a right group, then every $\mathcal L$-class is a $\mathcal H$-class and also a subgroup.

A semigroup $S$ is called right simple if it contains no proper right ideal. A semigroup that is right simple and left cancellative is called a right group. This is equivalent to saying that, for any ...
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definition of (semi)group (co)homology

I'm puzzled why "group cohomology" contains terms 'group' (instead of 'semigroup') and 'cohomology' (instead oh 'homology and cohomology'). I'm new to the subject. Please inform me of any claims ...
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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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42 views

Proof of Green's Relation

Can anyone tell me where I can get the proofs for the following Green's relations: aLb iff im(a) = im(b) aRb iff ker(a) = ker(b) aDb iff rank(a) = rank(b) Where a,b belongs to T3 transformation.