# 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.

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### Show that a semi-group (E,T) (I.e., with T associative) satisfying a certain property is a monoid (i.e., possesses a neutral element)

Let $E$ be a set with an internal operation $T$ associative such that there exist $a \in E$ such that : $(∀y\in E) (\exists x\in E) \ y=aTxTa$ Prove that $(E;T)$ has an identity element. What I have ...
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### Compactification of a topological semigroup with right-identity structure

I'm looking for a hint on this question: Let $T$ be as a semigroup with right-identity structure i.e. $rs=r$ for all $r, s\in T$. Also, we consider $T$ as a topological semigroup that is a locally ...
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### Regular semigroups -- intuition

I'm trying to develop some intuition around the definition of the pseudoinverse in a regular semigroup. Let the semigroup be $S$ with its associative operation written by juxtaposition. The ...
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### A semigroup with right identity and right inverses is a group.

Let $G$ be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an $e\in G$ such that $a· e = a$ for all $a\in G$. (b) Give $a\in G$, there exists an ...
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### Infimums of subsets of monoid.

I came up with the following idea This may have already been studied by someone else or may be a common sense idea in this field. Let $(M,+,\leq)$ be a totally ordered commutative monoid. For any non-...
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### $(ab)^2=a^2b^2$ doesn't hold for semigroup.

I am trying to prove that $(ab)^2=a^2b^2$ doesn't hold for a semigroup $S$. It is easy to prove(using inverse property) that the above condition holds for an abelian group. However, for semigroup, it ...
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### Are there theorems about extending a monoid without inverses to a group by reflecting the elements? [duplicate]

What I have in mind is something like the following: the natural numbers with addition form a monoid. You can imagine constructing the integers by taking the naturals, adding a set constructed by &...
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### Is there a monoid with universal property dual to that of a free monoid?

One of the important properties of a free monoid on a set is that for any other monoid, you can create a unique homomorphism from the free monoid to that monoid. Does there exist some monoid over a ...
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### Let $e$ be an idempotent of the monoid $M$, $x$, $y$ be two elements of $eMe$. Then, $(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$ if and only if $MxM = MyM$ [duplicate]
I am studying semigroups. I saw a Lemma in the text that states: Let $e$ be an idempotent of the monoid $M$, $x$, $y$ be two elements of $eMe$. Then, $(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$ if and only ...