Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 ...

2
votes
1answer
15 views

Strong Folner condition(SFC) implies the existence of a left Følner sequence.

I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says: Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left ...
0
votes
1answer
37 views

Finding an equivalence relation that isn't a congruence.

Let $B=S \times T$ be a rectangular band such that $|S|=|T|=3$. I've got to find an equivalence relation which is not a congruence in order to prove that at least one exists. I've tried many ...
1
vote
0answers
17 views

Does Wikipedia link from ‘quantale’ correctly? Residuated semigroups to residuated lattices.

From its article on quantales https://en.wikipedia.org/wiki/Quantale Wikipedia offers a link to ‘residuated semigroups’, but the link actually goes to an article on residuated lattices, https://en....
0
votes
3answers
35 views

How to distinguish between different elements in a semigroup?

I've recently started learning abstract algebra on my own and there is something that I'm struggling with at the moment. Suppose we have a semigroup with two elements $a,b$ and a binary ...
0
votes
1answer
113 views

Another Proof of Euclid's Theorem (infinite number of primes)?

Here $\mathbb N = \{2,3,4,\dots\}$ with the binary operation of addition. If $m \in \mathbb N$ we denote by $G_{\mathbb N} (m)$ the semigroup generated by $m$. Definition: A number $p$ is said to be ...
1
vote
0answers
21 views

Characterizing commutative semigroups with a factorization property.

Let $(N, \times)$ be a commutative semigroup and assume that a countably infinite subset $P$ of $N$ algebraically generates $N$, and let ${\mathcal F}(P)$ denote the set of all non-empty finite ...
3
votes
1answer
24 views

Induced Semigroup Structures via (left) Translation

Let $(M,\circ)$ be any semigroup satisfying the following properties: P-1: $\text{For every } x,y,z \in M \text{, if } z \circ x = z \circ y \, \text{ then } \, x = y$. If $\zeta \in M$ we define $...
4
votes
0answers
95 views

Subsemigroup of a finite semigroup

Let $S$ be a finite semigroup and $T \subseteq S$ which satisfy the following property: For $x, y \in T$, we have $x, y \in \langle z \rangle$ for some $z \in S$. If $H \subseteq S$ satisfy the above ...
0
votes
0answers
12 views

Proof that for a strongly continuous contraction resolvent, there is exactly one linear operator that generates the resolvent.

I have questions about the proof of the following Proposition from the book Introduction to the Theory of Non-Symmetric Dirichlet forms. First, how do we get the independence of $G_\alpha (B)$ of $\...
3
votes
1answer
66 views

On the size of a semigroup generated by 5x5 matrices

I was working on a problem with a friend where we were given two 5x5 matrices $A$ and $B$ with entries in $\{0, 1, -1\}$, which must generate a semigroup (under matrix multiplication) of order $>...
1
vote
1answer
36 views

Complex function with binary operation

I'm working on a question on complex function with binary operation, it has two parts in the question and I got stuck on part b. Part a: represent the $∘$ operation in a graphical way. I have drawn ...
1
vote
1answer
55 views

Exercise $1.9.3$ of Howie's “Fundamentals of Semigroup Theory” follow up

In the following link there is an answer to a question I am working on but I'm nut sure I understand it fully. Exercise 1.9.3 of Howie's “Fundamentals of Semigroup Theory”. The second question: ...
1
vote
1answer
25 views

Equivalence relation classes with ideals

I'm trying to understand some notation which is unfamiliar to me and I am struggling to see the logic behind it. I gather that the use of $[x]_\rho$ represents the set of $\rho$ equivalence classes ...
1
vote
1answer
32 views

Ideals of a Semigroup - Exercise 1.9.19 of Howie's “Fundamentals of Semigroup Theory”.

I am working on excercise 1.9.19 of Howie's “Fundamentals of Semigroup Theory”: Let I, J be ideals of a semigroup S. Show that I$\cap$J and I$\cup$J are ideals of S. I am really struggling with ...
0
votes
1answer
30 views

Consider the set F under the operation of composition of functions ◦.

Let $C = \{z \in \mathbb C \mid |z| = 1\}.$ Let $f_\theta : \mathbb C \to \mathbb C$ be given by $f_\theta (z) = e^{i\theta z}$. Let $F = \{f_\theta | \theta \in \mathbb R\}$. Consider the set $F$ ...
0
votes
1answer
53 views

Examples of Commutative Semigroups Where the Cardinality of the Carrier Set is Greater Than $\mathfrak c$.

Given: A set $M$. A binary operation $+$ defined on $M$ $+: M \times M \to M$ $\text{ that is both associative and commutative.}$ satisfying the following properties: P-1: $\text{For every } x,y,z \...
0
votes
1answer
45 views

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 ...
1
vote
1answer
46 views

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 ...
4
votes
2answers
110 views

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 ...
1
vote
1answer
43 views

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 ...
1
vote
1answer
45 views

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; ...
-2
votes
1answer
52 views

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?
1
vote
2answers
44 views

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 ...
2
votes
1answer
33 views

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. ...
5
votes
1answer
41 views

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 ...
6
votes
3answers
105 views

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 ...
3
votes
0answers
49 views

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 $...
1
vote
0answers
31 views

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 $\forall x,y \in S : axbayb = axyb$? ...
6
votes
1answer
95 views

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 ...
2
votes
0answers
19 views

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 \...
1
vote
0answers
36 views

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 ...
0
votes
3answers
39 views

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?
3
votes
1answer
73 views

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 ...
6
votes
3answers
224 views

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. ...
3
votes
1answer
62 views

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 ...
1
vote
0answers
39 views

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, \...
1
vote
1answer
47 views

(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 ...
0
votes
0answers
36 views

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 $\langle - 1,d \rangle \cong \langle -1, 1\rangle$...
0
votes
0answers
22 views

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 \...
2
votes
1answer
54 views

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, \...
0
votes
1answer
102 views

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 ...
1
vote
1answer
59 views

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 ...
0
votes
2answers
42 views

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 ...
6
votes
1answer
63 views

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, ...
1
vote
0answers
34 views

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=\...
1
vote
0answers
34 views

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 ...
10
votes
2answers
97 views

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 ...
1
vote
0answers
35 views

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)\...
2
votes
1answer
36 views

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,...
4
votes
0answers
55 views

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....