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. This tag is most frequently used for questions related to the concept of semigroups in the context of abstract algebra. Please use the more ...

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Reference request for tricky problem in elementary group theory

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 ...
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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 ...
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Prove that this 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 $b*a=c*a$, ...
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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?
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Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a 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? ...
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Help me prove equivalently of regular semigroup and group.

Let $S$ be a semigroup. Prove that the following are equivalent: $\forall a \in S \exists! x \in S$ such that $ax \in E(S)$ where $E(S)$ is the set of all idempotent. $\forall a \in S \exists! x \in ...
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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 ...
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1answer
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Why is a monoid with right identity and left inverse not necessarily a group? [duplicate]

This problem is from Herstein's 'Topics in Algebra'. I've thought about it a bit but haven't come up with much. Let $G$ be a non-empty set with an associative product which also satisfies: $\exists ...
2
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1answer
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Free idempotent semigroup with 3 generators

On Finite free objects I asked examples of finite free objects. I got answer "The free idempotent semigroup satisfies $x^2=x$ for each element $x$. And I confirm it is finite if it is finitely ...
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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 any ...
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Group identities and inverses

A set is a set. A magma is a set with a binary operator. A semigroup is a magma with an associative binary operator. A monoid has a two-sided identity. And a group has two-sided inverses. I am ...
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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 ...
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8answers
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Does a non-abelian semigroup without identity exist?

I was introduced to semigroups today and had a question. So all the examples of semigroups I was given were either monoids or groups. So I was curious, does there exist a semi-group which is not ...
3
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1answer
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Is every semigroup with (possibly non-unique) division a group?

Let's say that a semigroup $(S,\cdot)$ has weak division if for all $a,b\in S$ there exist $c,d\in S$ such that $ac = b$ and $da = b$. Note that we don't require $c$ and $d$ to be unique. This ...
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2answers
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Whether $L=\{(a^m,a^n)\}^*$ is regular or not?

I am condidering the automatic structure for Baumslag-Solitar semigroups. And I have a question. For any $m,n \in Z$, whether the set $L=\{(a^m,a^n)\}^*$ is regular or not. Here a set is regular means ...
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1answer
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Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
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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 ...
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Can we extend the definition of a homomorphism to binary relations?

This is going to be quite a long post. The actual questions will be at the end of it in section "Questions." INTRODUCTION After receiving an answer to this question about extending the definition of ...
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1answer
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How many associative binary operations there are on a finite set?

I am studying Scott's book Group Theory. In the Exercise $1.1.17$ he asks us to show that if $S$ is a set and $|S|=n$, then there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$. But ...
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6answers
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What are the subsemigroups of $(\mathbb N,+)?$

While trying to solve a somewhat bigger problem, I realized that I don't know what the subsemigroups of one of the most important semigroups, $(\mathbb N,+)$, are. (I assume $0\not\in\mathbb N$.) I've ...
10
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1answer
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Ring without distributive law?

I recently came across a binary operation (in a very non-algebraic context - it's a way to organize a certain updating of log-likelihood-ratios) and was idly wondering whether it is any kind of ...
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A cyclic subsemigroup of a semigroup S that is a group

I came across this problem while reading some lectures about semigroups here Lecture Notes on Semigroups by Tero Harju. Page $10$. He named it a nontrivial exercise. Let $r$ be the index and $p$ the ...
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If cancellation laws hold, then a finite semi-group is a group [duplicate]

Show that if both cancellation laws i.e $w.a = w.b \implies a = b$ and $a.w = b.w \implies a = b$ holds then a finite semi-group (a finite set with associative binary operation) is a group. I have ...
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3answers
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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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Seeking more information regarding the function $\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right].$

Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows. $$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$ The motivation is that according to ...
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Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?

This question is a by-product of this one. I'm asking it because of this comment by Tara B. I'll repeat the definitions. The full transformation semigroup $\mathscr T_X$ on a set $X$ is the semigroup ...
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1answer
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Prove that $\{S(t)\}_{t \ge 0}$ is a contration semigroup on $L^2(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the ...
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Name for a function whose image has smaller cardinality than its domain

I asked this question in the comments of this question, whose title would have done just as well for mine. But I suppose it should be a separate question. Is there a name for functions ...
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Is there a name for this property?

Is there a general name for the following properties, (similar to the properties of existence of an additive identity, existence of multiplicative identity etc): For any given set, the intersection ...
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1answer
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Nowhere commutative semigroups = rectangular bands?

I have recently read about Nowhere commutative semigroups, and there, they say, if $S$ is a semigroup, then these statements are equivalent: $S$ is nowhere commutative($ab=ba$ implies $a=b$). $S$ is ...
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1answer
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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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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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6answers
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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 ...
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Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like ...
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semidirect product of semigroups

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function ...
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1answer
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Semigroups and solutions of equation

It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ya=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ya=b$ has ...
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1answer
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Proving that if the semigroup (A, *) is a group, then the relation is an equivalence relation.

I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this: Throughout this ...
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1answer
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Weak generator of Feller semigroup

Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in ...
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1answer
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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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1answer
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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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1answer
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Structure Descriptions (GAP) in semigroups

As can be easily seen through GAP manual: 37.6 Structure Descriptions StructureDescription( G ) A The method for StructureDescription exhibits the ...
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How to move from a right semigroup action to a left semigroup action?

Let $S$ be a semigroup and $X$ any set. Define a left action of $S$ on $X$ to be a map $\sigma: S \times X \rightarrow X$ with the property that $(st)x = s(tx)$, where we define $gx = \sigma(g,x)$ ...
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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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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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1answer
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Does every continuous time minimal Markov chain have the Feller property?

Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative off-diagonal entries.) As explained for ...
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1answer
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Can a smooth function on the reals form a non-commutative semigroup?

Let $f\colon \mathbb{R}^2 \to\mathbb{R} $ be a smooth function. Can there exist an algebraic structure $(\mathbb{R}, \cdot)$ such that for $x,y \in \mathbb{R}$, $x \cdot y = f(x,y)$ that is a ...
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Evans' theorem of embeddings into 2-generator semigroup.

I recently found this theorem when I am studying Semigroup presentations. Theorem: If S is a semigroup generated by denumerably many elements, then S can be embedded into a semigroup generated by ...
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1answer
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Two elementary question on automaton and language

1.What is the definition for a semigroup(or monoid) recognizing a set of words(or language)?2.Are recognizable,rational and regular equivalent to each other with respect to a language? PS:The reason ...
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1answer
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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 ...
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Is there a topology on the full transformation semigroup?

$\mathscr T_X$ will denote the set of all functions from a non-empty set $X$ into itself, with the binary operation of composition $\circ$ making it a semigroup, called the full transformation ...