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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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 ...
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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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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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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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Proof of Green's Relation

Can anyone tell me where I can get the proofs for the following Green's relations? $a\mathcal{L}b$ iff $\operatorname{Im}(a) = \operatorname{Im}(b)$, $a\mathcal{R}b$ iff $\operatorname{ker}(a) = ...
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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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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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Semigroups (ideals of a semigroup)

How many ideals are there in the $\mathbb Z_{28}$? $\mathbb Z_{28}=\{0, 1, 2, 3, 4, 5, ..., 27\}$ is a semigroup under multiplication modulo 28.
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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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semigroup ideals

For the semigroup $$‎S_{3 \times 3} = \bigl\{(a_{ij}) \bigm| a_{ij} \in \mathbb Z_2 = \{0,1\}\bigr\}$$ ‎‎(the set of all $‎3‎\times‎‎3$ matrices with entries from $\mathbb Z_2$) under multiplication. ...
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Semigroups isomorphism [duplicate]

Does there exist an isomorphism between the semigroups $S(4)$ and ‎‎‎‎‎‎$\mathbf Z_{256‎‎‎‎‎‎‎}$.‎ $S(4)$ is the set of all maps from the set $X$ to itself and $X = \{1, 2, 3, 4\}$. $S(4)$ is a ...
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Semigroup isomorphism

Does there exist an isomorphism between the semigroups $S(4)$ and $\mathbb{‎‎‎‎‎Z}_{‎256}$?‎ $S(4)$ is the set of all maps from the set $X$ to itself and $X =\{1, 2, 3, 4\}$, $S(4)$ is a ...
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Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
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G is a group if and only if for all $a,b∈G$, $ax=b$ has solution; true or false? Why?

I think we only need $ax=b$ or $ya=b$ have solutions in $G$, I'll prove it. Proof: (I) $G$ is a group $\implies$ $a^{-1}ax=a^{-1}b$ $\implies$ $x=a^{-1}b$ $\implies$ $ax=b$ has solutions in $G$ ...
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Idempotents in $M_2(\mathbb{C})$

Given two idempotents $e,f\in M_2(\mathbb{C})\setminus\{I_2\}$, the sets $$\{eg^{-1}:g\in GL_2(\mathbb{C}), eg^{-1} \text{ is an idempotent}\}$$ and $$\{gf:g\in GL_2(\mathbb{C}), gf \text{ is an ...
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Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
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Why $f(x)=x$ for any $x\in f(X)$?

The set $S(X,X)$ of all mappings of a set $X$ to itself with the composition of mappings in the role of multiplication, where $|X|>1$. Why is not it a group? Let $X$ be a nonempty set. Then the ...
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Example for a claim

If $H$ is a subsemigroup of a semigroup $S$, then it may happen that, for some $a$ and $b$ in $S$, the sets $aH$ and $bH$ don't coincide and, nonetheless, are not disjoint. Does there exist an ...
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How could I understand this sentence:

How could I understand this sentence: For any subgroup $H$ of a group $G$ we have $H^2=HH=H$. However, in general, the same is not true for subsemigroups of semigroups. Thanks a lot!
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How can I show $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$ where $S(t)f(x)=e^{-t^2-2tx}f(x+t)$..

I need some help with the following: For every $t\in [0, \infty)$ let $S(t):C_0([0, \infty))\rightarrow C_0([0, \infty))$ be the bounded linear operator given by, $$S(t)f(x)=e^{-t^2-2tx}f(x+t)),$$ ...
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How could I prove that the equality $a^na^m=a^{n+m}$ in a semigroup with identity?

Let $S$ be a semigroup with a identity. How could I prove that the equality $a^na^m=a^{n+m}$ which holds for all $n,m \in \mathbb Z$. Note that $a \in S$ and $\mathbb Z$ denotes the set of all ...
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Is associative binary operator closed on this subset?

Here is the problem: Suppose that $*$ is an associative binary operation on a set $S$. Let $$H:= \{a \in S\mid a * x = x * a \mbox{ for all }x\in S\}.$$ In other words, $H$ is consisting of all ...
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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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How to show that $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$?

For every $t\in[0, \infty)$ consider the bounded linear operator $S(t):C_{ub}(\mathbb R)\rightarrow C_{ub}(\mathbb R)$ given by $$S(t)u(x)=u(x+t),$$ where $C_{ub}(\mathbb R)$ is the set of all bounded ...
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Why is two-element null semigroup excluded from the $0$-simple semigroup definition?

My question is probably a little bit silly, but still.. The definition of $0$-simple semigroup states, that a semigroup $S$ with zero is called $0$-simple, if $\{0\}$ and $S$ are it's only ideals and ...
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Confusing in periodic semigroup

Can anyone tell my, why we have to mod $r$ ? Thank you so much.
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uniqueness of solutions of $ax=b$ and $ya=b$ in a semigroup . [duplicate]

Suppose $G$ is a semigroup in which every equation of the form $ax=b$ or $ya=b$ has a solution. Does this solution have to be unique?
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Prove that any completely regular semigroup $S$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$

Consider any completely regular semigroup $S$. I would like to prove, that any $a,b\in{S}$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$. So far, I was able to prove only the first ...
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A statement about an element $a$ in semigroup S, such that $aS$ containts idempotent and $a=axa$ implies $x=xax$

I have been currently studying some characteristics of completely regular and completely simple semigroups and I have came across a lemma, which seems simple, but I'm struggling with it's proof, so I ...
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Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
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Is a direct product $\prod_{\alpha\in{}A}S_\alpha$ of semigroups $S_\alpha$ simple, if all semigroups $S_\alpha$ are simple?

I am currecntly trying to give an answer to the following problem. Consider a family of semigroups $(S_\alpha)_{\alpha\in{}A}$ and let every semigroup $S_\alpha$ be simple. Is it true or not, that ...
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Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
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The $\mathcal{J}$- class of a primitive idempotent in a regular semigroup.

I am currently studying some basic facts of regular semigroups and Green's relations and I got stuck on the following exercise problem. Let $S$ be a regular semigroup with a primitive idempotent $e$. ...
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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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Maximal ideals in rings

Let $R$ be a ring with identity. Is it true that if $R$ has a finite maximal right ideal then it MUST have a finite maximal two-sided ideal ?