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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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$A$ is a subsemigroup of $S$ if and only if $A^{2}\subset A$

Question: Show that $A$ is a subsemigroup of $S$ if and only if $A^{2}\subset A$. The subset here may not necessarily be proper. My approach, Suppose $A$ is a subsemigroup of $S$, then for all $x,x\in A, x^{2}\in A$. Does this mean that,…
Hassan Muhammad
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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 $\operatorname{ker}(a) = \operatorname{ker}(b)$, $a\mathcal{D}b$ iff…
QueenLA
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Confusing in periodic semigroup

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

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?
SE Anarchist
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Examination of idempotent elements of the Catalan monoid

Let $\mathcal{C}_n$ be the Catalan monoid of self-maps, that is, $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(x)\le x$). For any semigroup…
1ENİGMA1
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semigroup presentation and Diamond lemma

Suppose a semigroup (possibly infinite) presentation is given with generating set $S$ and relations $R$. I need to prove using Bergman's diamond lemma that the semigroup is non-zero i.e, I have to give normal forms of elements of the semigroup.…
user73789
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is $a \circ b \circ a=a$ when $[A,\circ]$ is a semi-group

Let $[A,\circ]$ be a semi-group. Further $\forall a,b$ if $a \neq b$, then $a\circ b \neq b \circ a$. Now show that $a\circ b\circ a=a$ My argument is as follows: since in semi-group we have Associativity property, So, we have $(a\circ b)\circ a=a…
Aayush Upadhyaya
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How do the Green's R and L Relations relate with left and right Coset?

How do the Green's R and L Relations relate with left and right Coset? Are there any comparisons? I do know that both tools are used to partition algebraic structures into disjoint substructures. I actually did some paper in characterising…
Marshal Sampson
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order of subsemigroups

For every finite group $G$, if $H$ is a subgroup of $G$, then the order of $H$ divides the order of $G$. I know that if $S$ is a semigroup, by adjoining an identity element to $S$, clearly $S$ is a subsemigroup of $S^1$. So in general, the order of…
khers
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Find the Automorphism group of a Brandt semigroup $B(G,2)$ ,where $G $ is a cyclic group of order 4

Find the Automorphism group of a Brandt semigroup $B(G,2)$ ,where $G $ is a cyclic group of order 4. Take $G=\{ e, a ,a^2, a^3\}$ $B(G,2) = \{ (i,a^s , j) : 1 \leq i,j \leq 2 \ \ , 0\leq s \leq 3 \} \cup \{0\}$ and the binary operation is…
user120386
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Relation between $\mathcal R$ , $\mathcal L$ , $\mathcal H$ and $\mathcal D$ class on a Semigroup $S$

I am studying Semigroup Theory from a book by John M. Howie. I have questions about THE STRUCTURE OF $\mathcal D -$ CLASSES on page 49: If D is an arbitrary $\mathcal D - $ class in a Semigroup $S$ and if $a, b \in D$ are such that $a \mathcal R b$…
Struggler
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Do endormorphisms of a semigroup form a monoid? Automorphisms form a group?

I came across this theorem when studying thess lecture notes Theorem: Prove that The endomorphisms of a semigroup S form a monoid. The automorphisms of a semigroup S form a group. I do not have even a small idea of how to construct the proof.…
Hassan Muhammad
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The set of natural number is semi group?

Because first two properties of group are satisfied but I have read in local author book it is not semi-group.
Navid Gujjar
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