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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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Classify monoids that are generated by one element.

Algebra by Michael Artin Exer 2.M.4 M.4. A semigroup S is a set with an associative law of composition and with an identity. Elements are not required to have inverses, and the Cancellation Law ...
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Can the set of odd primes be decomposed into $\Bbb{P} = A + B, $ for some $A,B \subset \Bbb{Z}$?

Can there ever exist infinite sets of integers $A, B$ such that $A + B = \{ a + b: a \in A, b \in B\} = \Bbb{P}$? Where $\Bbb{P}$ is the set of odd primes? You can include $0$ and / or $\pm$ odd ...
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Commutative Semigroup

Let $S$ be a Semigroup with the two following properties, $(1):$ for all $x$ in $S$ we have $x^3=x$ $(2):$ for any $x,y$ in $S$ we have $xy^2x=yx^2y$. Then prove that this Semigroup $S$ is ...
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Check the properties of the following operation defined on R

An operation is defined on $\mathbb{R}$ such that for every $x,y \in \mathbb{R}$, $x \ast y=\sqrt{x^2+y^2}$. I was checked some of the basic properties like commutativity, associativity and whether ...
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Isn't having both assumptions that $ax=b$ and $ya=b$ have solutions $\forall a,b \in G$ redundant?

In the book of Algebra by Hungerford, at page 25, it is given that However, in the proposition 1.4, the one of the conditions that $ax =b$ and $ya=b$ have solutions $\forall a,b \in G$ is redundant; ...
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Determining domain of differential operators in a concrete semigroup problem

Let s>3/2, $H^s=H^s(\mathbb{R})$ be the Sobolev space of order $s$, $B$ be the set of bounded operators from $H^{1/2}$ to itself, $u\in H^s$ and $A(u):=u\partial_x$ an operator. How can I determine ...
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Question on proving $γ∨ρ=γ∘ρ∘γ$

in theorem 5 of Group congruences on eventually regular semigroups by S. Hanumantha Rao. he saied it suffices to prove $ρ∘γ∘ρ⊆γ∘ρ∘γ$. Why?
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Suppose $G$ is a semigroup and it holds both left and right cancellation.

Suppose $G$ is a semigroup and it holds both left and right cancellation. Also for each $a,b\in G$, $xa=b$ has solution in $G$. Prove G is a group. I know this question looks very "old" style. First ...
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Understanding a particular group/semigroup operation

Let $\odot$ be the binary operation defined by $$ x\odot y := (x+y)+(x\cdot y)$$ where $+$ and $\cdot$ are the usual operations of addition and multiplication from whatever ring you're working with. ...
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Heat semigroup equality $e^{t\Delta}p(x,x_0, 1)=p(x,x_0,1+t)$ ?

Let $\Omega$ be a domain in $\mathbb{R}^n$ and consider the heat semigroup $e^{t\Delta}$ of $\Omega$ where $\Delta$ denoted the Laplacian on $\Omega$ with Dirichlet boundary condition. Let also $p(x,y,...
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Implicit operations in finite semigroups.

what are some examples of implicit operations in finite semigroups other than expressions involving $\omega$? Like $x^\omega y^\omega$ or $x^{\omega+1}$. By Reiterman's theorem, pseudovarieties of ...
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Is a monoid commutative if $(ab)^2=a^2b^2$?

Let M be a monoid. Suppose that: $(ab)^2=a^2b^2$ for any elements a,b in M. Is M commutative? The result is obviously true for groups, but I can't find a counterexample for monoids. And without ...
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Factorising a divisor of a product

In the ring of integers (or the monoid of natural numbers under multiplication), I believe that the following theorem holds: Lemma Set $m$, $a$, $b$. If $m | ab$ then there exist $u$, $v$ such that $...
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Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$?

Based on J.-E. Pin's answer here, I'd like to know the following: Question. Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$? ...
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Terminology for a “subgroup” that has a different identity element.

Let $M$ denote a monoid. Then to refer to submonoids of $M$ that just happens to be a group, I think the phrase "subgroup of $M$" is okay, as it's unlikely to cause confusion as long as you instruct ...
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Direct sum of reals and quaternions is not a semigroup algebra for some semigroup

For a given semigroup $S$, the semigroup algebra over some field $F$ is the set of formal sum with the convolution product, and is denoted by $F[S]$. If we built the direct sum $$ \mathbb R \oplus \...
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Direct sum of Matrix algebras is not isomorphic to some semigroup algebra

An $n \times n$ matrix unit is any matrix which has zeros every, except at one position where it has one. By $E_{ij}^{(n)}$ we denote the $n \times n$ matrix unit which has its one at the $i$-th row ...
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Proving a left identity [closed]

I have the following question: I know that I need to prove that any element with a+b=1 multiplied by S will yield S. Can anyone give me a starting point for this proof?
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Fractional Brownian Motion and Fractional Laplacian

It is well known that the Laplacian is the infinitesimal generator of a Brownian Motion, that is, $$ \lim_{t \to 0} \frac{E[f(x+B_t)-f(x)]}{t}= \Delta f(x). $$ Is it true that for the Fractional ...
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semilinear parabolic pde-reference

i have the following semilinear parabolic problem $$ \partial_t u(x,t)- \Delta u +F(u)= f(x,t); x \in \mathbb{R}^n, t > 0 $$ $$ u(x,0)=0 $$ with periodic boundary conditions, and $F$ is non linear. ...
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In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column?

In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column? It's obviously not the Closure axiom, and after some inspection, I believe the Inverses axiom does fail. ...
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Expressing Green's relations in regular semigroups

Let $S$ be a semigroup and $a \in S$. An element $a' \in S$ is called an inverse of $a$ if $$ aa'a = a \qquad a'aa' = a'. $$ Denote the set of all inverses by $V(a)$. A semigroup where every ...
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About 3-dimensional quadratic space

3.3 Theorem. Assume that every $3$-dimensional quadratic space over $K$ is isotropic. Let $\phi$ be a regular $n$-dimensional quadratic space. Then $$ \phi \cong \langle \delta, 1, \...
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(Corrected in Wikipedia) 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 the group law on the extended square class group

We define a group law on the set $$ Q(K) := \mathbb{Z}/2\mathbb{Z} \times K^\bullet/K^{\bullet 2} $$ as follows \begin{align*} (0, \alpha) + (0, \beta) &= (0, \alpha \beta) \\ (1, \...
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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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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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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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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 ...