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.

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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Evan's 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 denummerably many elements, then S can be embedded into a semigroup generated by ...
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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 offdiagonal entries). As explained for ...
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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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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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About the multiplicity of a semigroup ring.

Let $A=K[X^{n_1}, \dots, X^{n_s}]$, $S=\langle n_1, \dots, n_s \rangle$ ($n_1 < \cdots < n_s$) a numerical semigroup and let $\mathfrak m$ be the maximal ideal $(X^{n_1}, \dots, X^{n_s})$. We ...
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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 ...
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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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Errata for Semigroups and Combinatorial Applications by G. Lallement?

I am asking this question after years of frustration with the typos in the subject book I have read. It has been cited and referenced by many literatures and books in math and computer science. ...
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under what conditions a product of matrices is the identity matrix (more complicated than that)?

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple ...
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Existence of universal enveloping inverse semigroup (similar to “Grothendieck group”)

Context In its simplest form, the Grothendieck group construction associates an abelian group to a commutative semigroup in a "universal way". Now I'm interested in the following nilpotent ...
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Is there a special name for a semigroup whose multiplication is a constant function?

Let $S$ be a (commutative) semigroup with distinguished element 0 such that $ab=0$ for $a,b\in S.$ Of course this is a very simple family of semigroups, defined only by their cardinality. Does it ...