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Questions tagged [semigroups]

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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Every subsemigroup is in its own normalizer

Let $G$ be a group and $S \subseteq G$. We define the normalizer of $S$ as $N(S) := \{ n \in G : nS = Sn\}$ According to Wikipedia; If $S$ is a subsemigroup of $G$, then $N(S)$ contains $S$. But ...
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Isometry of quadratic space

Every element of $K$ is a square if and only if every 2-dimensional form over $K$ is isotropic. In fact, if $\langle -1, d \rangle$ is isotropic, then $\langle - 1,d \rangle \cong \langle -1, 1\rangle$...
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Show that ordinal product of two semilattices, one of them uniform, is subuniform

A semilattice $U$ is called uniform, if for every $x, y \in U$ we have for the principal ideals $Ux \cong Uy$. A semilattice is called subuniform if $$ \forall x,y \in U \exists z \in U : z \le y \...
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about group law on extended square class group

This is excerpt from W. Scharlau: Quadratic and Hermitian Forms, page 37. In this $Q(K)$ we define group law as given I don't understand what is third law saying (1,$\alpha$)+(1,$\beta$)=(0,-$\alpha$$...
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About Semigroup & Semiring homomorphism

This is Theorem 2.1.1 from Scharlau's book Quadratic and Hermitian Forms Can somebody explain me why $R$ has zero element $[a,a]$ and negative element $[b,a]$? My guess: I think it is due to the ...
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1answer
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Prove that there is no bijective homomorphism from $\left(\mathbb{Q},\ +\right)$ to $\left(\mathbb{Q_+^*},\ \times \right)$

I need to prove that there does not exist any bijective homomorphism from $\left(\mathbb{Q},\ +\right)$ to $\left(\mathbb{Q_+^*},\ \times \right)$ Here is a way to prove it: Let $f$ be a ...
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2answers
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Example of a semigroup satisfy some properties

I want to find an example a finite semigroup $S$ and $K \subseteq S$ satisfy the properties For any $a,b \in K$, we have $a,b \in \langle c \rangle$ for some $c \in S$. $K$ does not hold closure ...
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Show that a semigroup is semisimple iff $A^2 = A$ for every two-sided ideal $A$

Let $S$ be a semigroup, a subset $I\subseteq S$ is called an ideal if $SI \subseteq I$ and $IS \subseteq S$. We denote by $S^1$ the semigroup $S$ adjoined with a identity if it does not contains one, ...
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isomorphism generalized semigroup

I would you like to construct an isomorphism with generalized full transformations. All case for finite sets. $\textbf{Case I}:$ Let $\theta:Y\rightarrow X_n$ is a bijection. Then $S=\...
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Semigroup of differentiable functions on real line

Let $D(\mathbb R) $ be the set of all differentiable functions $f: \mathbb R \to \mathbb R$. Then obviously $D(\mathbb R)$ forms a semigroup under usual function composition. Can we characterize (upto ...
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Show that a semigroup with $aS \cup \{a\} = bS \cup \{b\}$ and $Sa \cup \{a\} = Sb \cup \{b\}$ is a group

Let $S$ be a semigroup such that for all $a,b \in S$ $$ aS \cup \{a\} = bS \cup \{b\} \quad \text{and} \quad Sa \cup \{a\} = Sb \cup \{b\}. $$ where $aS = \{as : s \in S \}$ and similarly $Sa$. I ...
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Linear transformations with restricted range

Let $W$ be a subspace of a vector space $V$ over a finite field $F$ and let $L(V,W)$ denote the set of all linear transformations from $V$ into $W$. Let $f$ be an element in $L(V,W)$ with $f(V)\...
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1answer
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Groups in the generalized triple “semidirect” product of semigroups

A semigroup $S$ acts on another semigroup $V$ (written additively for better readability, but could be non-commutative) on the left if $$ s(v_1 + v_2) = sv_1 + s v_2, \quad s(s')v = (ss')v $$ for $s,...
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Constructing free monoid from free semigroup

Given a monoidal category $(\mathcal{C}, \otimes, I)$ with coproducts, the free monoid on an object $A \in \mathcal{C}$ is usually constructed by first constructing the free pointed object on $A$, i.e....
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make me idempotent!

$T_n$ be the full transformation semigroup on $X_n= \{1, 2, \cdots , n\}$. $D_r =\{\alpha \in T_n: |im(\alpha)|=r\}$. $E(D_r)$ is the set of all idempotents of semigroup $T_n$. $shift(\alpha)=\{x: ...
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Inclusion relations between equationally defined classes of finite semigroups

Let $S, T$ be two semigroups. In the following all semigroups are supposed to be finite. We write $S \prec T$ if there exists a surjective semigroup morphism from a subsemigroup of $T$ onto $S$. A ...
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Question on proof related to inclusion relations of principal ideals generated by elements.

Let $S$ be a semigroup, then an ideal $W$ is a subset $W \subseteq S$ such that $sWt \in W$ for all $s,t \in S^{\bullet}$, where $S^{\bullet}$ is $S$ with an additional unit element added if $S$ does ...
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1answer
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Does every submonoid of $\mathbb N_{\ge 0}$ contained in some numerical semigroup?

Let $A$ be a submonoid of the monoid of non-negative integers $\mathbb N$ under addition. Then does there necessarily exist a submonoid $S$ of $\mathbb N$ such that $A\subseteq S$ and $\mathbb N \...
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cosets in a semigroup

I have something like this: Let us consider a semi-group diagonal homomorphism $\Delta: G\rightarrow G\times G$ and let $H$ be the set of cosets of $\Delta(G)\in G\times G$. Then $H$ is a quotient ...
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Partially-ordered and (semi-)lattice-ordered semigroups and monoids

I'm interested in expository material, in the form of books, chapters in books, articles, blogposts, etc., about partially-ordered, and lattice-ordered semigroups and monoids. I have found two ...
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The unit group as a homomorphic image in a semigroup

In this paper, the author states in the first sentence: Among the homomorphic images of a semigroup (= a set closed with respect to an associative binary operation) there is at least one group, ...
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1answer
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Semigroup epimorphism that is not a quotient by a subsemigroup

It is not hard to see that if we have a semigroup epimorphism $\varphi\colon S\to T$, where $S$ is a group, then $T$ must be a group, and it is actually the quotient by the kernel, i.e. the set of $g\...
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Schreier varieties and Markov properties

Let $P$ be a property of finitely presented algebra preserved under isomorphism. $P$ is called Markov if there exists a finitely presented algebra $L_1$ satisfying $P$ and a finitely presented algebra ...
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2answers
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Example of a semigroup such that it contains a sub-semigroup which is a non-trivial group

Give an example of a semi-group such that it contains a sub-semigroup which is a non-trivial group. I am not very sure what a sub-semigroup is but according to Wikipedia: The semigroup operation ...
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What can we conclude about $a$ if $a^2=a$ in a monoid or semigroup?

I found the following question in a test paper: Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say about $a$? Monoids are associative and have an identity element. ...
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1answer
77 views

Equivalent condition on finite semigroup

Let $S$ be a finite semigroup. An element $a$ is said to be maximal if for any $b \in S$ with $\langle a \rangle \subseteq \langle b \rangle$ we have either $a = b$ or $\langle b \rangle = S$. Since $...
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1answer
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Proving an alternative version of Lallement's Lemma

Let $\phi:S\rightarrow T$ be a (homo)morphism from a regular semigroup $S$ into a semigroup $T$. Then $\textrm{im}(\phi)$ is regular. If $f$ is an idempotent in $\textrm{im}(\phi)$ then there ...
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Is this structure a group?

I'm currently working on a problem where a nonempty set $S$ has $\cdot$ an associative and cancellable operation. It is a bit ambiguous but when it says operation I feel the problem is talking about a ...
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What is funny about monoids? [closed]

I was looking for an introduction to Semigroups and found an Introduction to Semigroups and Monoids by Pete L. Clark. The abstract says: I know many working mathematicians who let loose a smirk or ...
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If $\left(E, \cdot\right)$ is a commutative semigroup of idempotents, then $\left(E,\leq\right)$ is an upper semilattice.

Suppose that $\left(E, \cdot\right)$ is a commutative semigroup of idempotents then the relation $\leq$ on $E$ defined by $$a\leq b \iff ab =b,$$ is a partial order on $E$. Furthermore, $\left(E,\leq\...
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1answer
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Structure of finite commutative semigroup generated by two element.

We know that a finite abelian group generated by two elements is isomorphic to $C_r \times C_s$, where $C_r$ and $C_s$ are cyclic group of order $r$ and $s$. I am interested to know what is the ...
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1answer
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Uniqueness of two side zeroes of binary operation

I came across the following fact in group theory: Two-sided identity of binary operation is unique. Does the similar statement for two sided zero also holds? : Two-sided zero of binary ...
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Term for elements of a monoid in a corresponding monid ring

What would be a proper term to use to call elements of a monoid $M$ in a corresponding monoid ring $RM$ (where $R$ is a ring) or in a monoid algebra $kM$ (where $k$ is a field)? Calling them ...
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Terminology: Semigroups, only their “binary operations” aren't closed.

Motivation: Consider $\mathcal{X}=(X, +)$, where $X=\{-1, 0, 1\}$ and $+$ is standard addition. Then $\mathcal{X}$ is associative (where defined) but not closed. NB: There is an identity element in $...
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1answer
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Example of a quotient semigroup which can not be embedded into the finite semigroup

For $S=(S,\cdot)$ a finite semigroup and $\sim$ a congruence relation on $S$, we have a quotient semigroup $(S/{\sim}, \cdot)$ with the operation as $[s_1]\cdot[s_2]=[s_1\cdot s_2]$. $S \rightarrow S/{...
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Free Nilpotent Semigroup

Let a nilpotent semigroup be defined as in Neumann and Taylor's Subsemigroups of Nilpotent Groups. More precisely, we call a semigroup nilpotent of class n if it satisfies the following identity: $q_n(...
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1answer
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Properties of infinite semigroup

Let $S$ be a semigroup. An element $a \in S$ is said to be maximal if $\langle a \rangle \subseteq \langle b \rangle \subseteq S $ implies either $\langle a \rangle =\langle b \rangle $ or $\langle b ...
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1answer
31 views

Direct product of two finite monogenic semigroup

Let $S_1 = \langle a \rangle = M(m_1, r_1) , S_2 = \langle b \rangle = M(m_2, r_2)$ are two finite monogenic semigroup, where $m_1, m_2$ are the index of $S_1,$ $S_2$ and $r_1, r_2$ are the period of ...
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1answer
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Hypersubstitution, m-ary terms, semigroups, equivalent definitions

A hypersubstitution $\sigma$ is (see, for example, Universal Algebra and Applications in Theoretical Computer Science, by Denecke and Wismath) mapping from term $f_i(x_1,...,x_{n_i})$ to the term $\...
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Maximal subgroup of a finite semigroup (GAP)

It is a very basic notion in semigroup theory that $H_e$ ,i.e. the groups of units of $eSe$, is just the maximal subgroup of semigroup $S$ where $e\in E(S)$. Here, I want to find it by using GAP for a ...
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1answer
63 views

Hyperidentity, semigroups, bands.

Let a semigroup satisfy $F(x,x)\approx x$, where $F$ is a binary operation symbol.Let $B$ satisfy $x(yz)\approx (xy)z$ and $xx\approx x$. Does $B$ satisfy $F$ as a hyperidentity?We need only consider ...
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1answer
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Semigorup variety, hyperassociativity,idempotentunclear proof of $x^4\approx x^2$

Let $V$ be a hyperasociative semigroup variety. For hyperasociativiy see below. Then $V$ satisfies the following identity: $$x^2 \approx x^4.$$ A proof attempt is given here: If $V$ is idempotent (i.e....
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1answer
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a question on Ellis semigroup

In transformation group $($$X$,$G$$)$ where $X$ is compact and hausdorff suppose that for any $x$ $\in$ $X$ , $xG$ is finite. we can see that : $xG$$=$ $\overline {xG}$$=$$x$$E(X,G)$ {$xE(X,G)$ $|$ $...
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1answer
55 views

Isomorphism between finitely generated semigroups

May I ask you to make light on the following point. I know that there maybe certain conditions of the type of two semigroups. I simply looking articles or a nice hint-answer just for sure: Does ...
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1answer
39 views

Question about semigroups of permutations

Consider a semigroup $A\subseteq Sym(X)$. For $a,b\in A$, say that $a\leq b$ iff $b=ac$ for some $c\in A$. Suppose that $\leq$ is a linear order on $A$. Does it follow that $A$ is abelian?
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What concept does a natural transformation between two functors between two monoids viewed as categories correspond to? [duplicate]

Let $\mathcal{M}$ be the monoid $M$ viewed as a category (single object, arrows are elements of $M$) and similarly for $\mathcal{N}$ and N. Let $F : M \to N$, and $G : M \to N$ be two monoid ...
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Singleton alphabet smallest grammar problem. Would this be easier work with?

I want to prove that for strings $s$ over a singleton alphabet $\Sigma = \{a\}$, if $s = a^{x\cdot y}, \ x,y \neq 1$, then there exists a smallest grammar derived from the grammar: $$ S \to A^x \\ A \...
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1answer
102 views

Showing that a completely regular semigroup $S$ is orthodox if and only if $\left(\forall x,y \in S\right) \space xy=xyy^{-1}x^{-1}xy$.

From J. M. Howie's Fundamentals of Semigroup Theory, Exercise 9, page 139: A semigroup $S$ is called orthodox if it is regular and the idempotents form a subsemigroup. How would I show that a ...
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1answer
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How can i understant a free semigroup on a set from a general point of view

We have (A, ∗) as a semigroup and the set M.We can say that (A, ∗) is a free semigroup on set M if there is a function f: M → A such that for any semigroup (B,⊗) and any function g : M → B there is a ...
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1answer
62 views

Show that a congruence on the semigroup $S$ is 'minimal'

I am working through the exercises in Howie's 'Fundamentals of Semigroup Theory'. The following is from page 138: Let $S$ be a commutative semigroup, and let $a,b \in S$. Write $a|b$ if $\exists x \...