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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 semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of composition, not known to have an identity or inverses.…
Lubin
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Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I'm not entirely certain…
user23211
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Are there any interesting semigroups that aren't monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there are any interesting examples of semigroups $(X, \ast)$ for…
bryn
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Right identity and Right inverse in a semigroup imply it is a group

Let $(G, *)$ be a semigroup. Suppose $ \exists e \in G$ such that $\forall a \in G,\ ae = a$; $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$. How can we prove that $(G,*)$ is a group?
Mohan
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What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the structure of $M(G)?$ Is there any known example of…
user23211
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Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a non-empty finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? Theorem 2.2.1. [R. Ellis] Let $S$ be a…
user59671
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How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything nicer in quite a spectacular fashion, e.g. it is…
Jakub Konieczny
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A finite, cancellative semigroup is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, then $b=c$. I want to prove that $G$ is a group,…
user39794
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(Organic) Chemistry for Mathematicians

Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do with physical chemistry: cond-mat, fluids, QM of…
isomorphismes
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Do these “ultraweak” one-sided group axioms guarantee a group?

This post shows that the “left” group axioms, which only guarantee a left-identity and left-inverses, are sufficient to guarantee that a semigroup is a group. The same idea could be used to show that the “right” group axioms are also sufficient.…
WillG
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Is a semigroup $G$ with left identity and right inverses a group?

Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group? Now, If both the identity and the inverse are of the same side, this is simple. For,…
providence
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How to get a group from a semigroup

I am sorry if my question is too simple. Is every semigroup associated to a group? If no, what conditions should be satisfied for a semigroup to have an associated group? If yes, how can I find the group? I thought of the universal property. Let $X$…
ShinyaSakai
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When a semigroup can be embedded into a group

Under what assumptions can a semigroup $(S,*)$ be embedded into a group?
Martin Sleziak
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Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be one of those things a lot of people know about (and…
Shaun
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Eilenberg's rational hierarchy of nonrational automata & languages

In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised a Volume C dealing with "a hierarchy (called the rational hierarchy) of the nonrational phenomena... using rational…
David Lewis
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