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Questions tagged [semigroups]

A semigroup is an algebraic structure consisting of a set together with a single 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/universal algebra. Please use the more specific tag (semigroup-of-operators) whenever appropriate.

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, and semigroups with the notion of inverses are known as regular semigroups and inverse semigroups.

Semigroups are used in various areas of mathematics. $C_0$-semigroups are important in partial differential equations. Semigroups have also connections to automata theory.

Topological (and left/right topological) semigroups are also studied. Perhaps the best know result in this area is Ellis-Numakura lemma. Using Ellis-Numakura lemma, existence of idempotent ultrafilters can be shown.

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A completely simple semigroups with cancelaltions is a group (simple proof)

Is there a simple proof of the following fact: Fact. Let $S$ be a completely simple semigroup with cancellations, i.e. each of the equalities $xa=xb$, $ax=bx$ implies $a=b$. Prove that $S$ is a group. Using Sushkevich-Rees Theorem, I can prove it,…
Artem
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Showing that (T,*) is a semigroup where T is the trace and * is defined inside

Let $D$ be a $D−class$ of a semigroup $S$. The Trace of $D$ is $T = D ∪ {0}$ where $0$ is a symbol not in $D$. Define a binary operation $∗$ on $D$ by: $a ∗ b = ab$ for $a, b ∈ D$ and $ ab ∈ R_a ∩ L_b$ $a*b = 0$ otherwise Show that $(T, ∗)$ is a…
user2973447
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Show that no $\mathcal H$ class contains more than one inverse of $a$, where $\mathcal H$ denotes Green's relation

use the results : Let $a$ be an element of a regular $\mathcal D-$ class $D$ in a semigroup $S$. Then If $a'$ is the inverse of $a$, then $a' \in D$ and the two $\mathcal H$ - classes $\mathcal R_a \cap \mathcal L_{a'}$ and $\mathcal L_a \cap…
Struggler
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Show that a semigroup $S$ is a rectangular band if and only if $ab=ba \Rightarrow a=b$

Show that a semigroup $S$ is a rectangular band if and only if $ab=ba \Rightarrow a=b$. (For all $a,b\in S$) I have the definition of a rectangular band as $aba=a$. When I try to prove this I keep getting stuck. This is my best effort.…
user2973447
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Proof of the theorem on numerical semigroups

$S$ - numerical semigroup, generated by $a$, $b$. $S=(a,b)={s0 < s1 < s2 < .....}$ $c$ - conductor of numerical semigroup: $c=(a-1)(b-1)$. All elements $x_i >= c$ are incremented by one. How can I prove it? My teacher wrote in (see pictures). Which…
Maria
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Finding representation in numerical semigroup

I'm given $(n_1,n_2,n_3)$, with $\operatorname{gcd}(n_1,n_2,n_3)=1$. Then, I need to find $c_1$, the least positive integer such that $c_1n_1=n_2\mathbb{N}+n_3\mathbb{N}$. I additionally need the specific coefficients $r_{12}$ and $r_{13}$ such…
Mark Schultz-Wu
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Find identity element, invertible and inverses in $T=\mathbb Z \times \mathbb Q$

Let the following operation be defined on $T=\mathbb Z \times \mathbb Q$: $$\begin{aligned}(a,b)\centerdot (c,d) = (-ac, b+d+2) \end{aligned}$$ in the commutative semigroup $(T, \centerdot)$, find the identity element, invertible elements and…
haunted85
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Greatest Common Divisor Semigroups

I'm trying to find a proof that shows if $a$, $b$ are in the natural numbers, then the sum of the additive semigroups $\mathbb N a + \mathbb N b$ is a subset of $\mathbb N d$ where $d = \gcd(a,b)$.
user130272
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Nilpotent Elements In Partial Maps Semigroup

Let $X$ be a set. The full transformation semigroup $T_X$ is the set of all maps from $X$ into $X$ with composition of maps as binary operation. A map $\alpha \in T_X$, is a subset of the cartesian product $X \times X$. A partial map $\beta$ is a…
Nicolas
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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 $x,y \in S$. Let $v$ be an inverse of $xyxy$, ie, $v=vxyxyv$…
Leafar
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The intersection of *-semigroups with I-semigroups is the class of inverse semigroups?

Howie in his Fundamentals of Semigroup Theory, 2nd ed., p. 103 writes The class of U-semigroups for which the unary operation satisfies the conditions both for a *-semigroup and for an I-semigroup is in fact the class of inverse semigroups, which…
the gods from engineering
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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 inverse semigroup $S$, then $\operatorname{tr}(\rho)=\{…
Hassan Muhammad
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$\mathcal{R}$-relation in full transformation semigroup

Let $T_{X}$ be the full transformation semigroup on $X$. For $\alpha$, $\beta \in T_{X}$ $$\alpha \mathcal{R}\beta \text { if and only if there exist }\gamma,\gamma' \in T_{X}:\alpha\gamma=\beta\gamma' .$$ This question that looks trivial, takes…
Hassan Muhammad
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How to show that S/J is a chain where S is a semigroup with no proper generating sets

I'm Struggling with exercise 4.7.17 of Howies Fundamentals of Semigroup theory Let $S$ be a non-trivial semigroup with the property that no proper subset of $S$ generates $S$. Show that each J−class of $S$ is a left zero semigroup or a right zero…
user207110
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Is this a*b = a is semigroup and commutative?

Let S be any nonempty set with the operation a * b = a. Is (S,*) a semigroup? Is it commutative? I dont know what to do if a * b = merely a. Usually if a * b = anything that have operation i know how to do. Can anyone show the proof as well? Thanks.
jackporter
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