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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 specific tag (semigroup-of-operators) whenever appropriate.

12
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2answers
318 views

Semigroups with no morphisms between them

Given two monoids we always have a morphism from one to the other thanks to the presence of the identity element. Are there examples of non-empty semigroups that have no morphisms from one to the ...
1
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1answer
45 views

Equal nr of $\mathscr D$-classes, different nr of idempotents

Are there examples of (finite) semigroups $S$ and $T$ such that they have the same 'number' of $\mathscr D$-classes, $S$ has idempotents and $T$ doesn't? Alternatively, they both may have idempotents, ...
1
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0answers
24 views

Semigroup and their monogenic semigroup

A subsemigroup $K$ of $S$ is said to be monogenic if $K = \langle a \rangle$ for some $a \in S$ and the order of $a$ is the size of $K$. If the order of $a$ is finite, then after some time power of $a$...
1
vote
1answer
20 views

Question about a near $\Bbb{Z}$-semimodule that has trivial addition.

https://en.wikipedia.org/wiki/Semimodule Suppose that we have a structure $M$ that has: Closure under addition. Closure under multiplication by any $a \in\Bbb{Z}$. The addition of any two elements ...
4
votes
1answer
54 views

Does every commutative idempotent semigroup have a representation as a union-closed family of sets

Consider a finite semigroup $S$ whose semigroup operation $\times$ is commutative and whose elements are idempotent. Does there exist a finite union-closed family of finite sets $\mathcal{M}$ such ...
2
votes
0answers
62 views

$G$ a set. When does a pair of subgroups of $\operatorname{Sym}(G)$ define “some” left and right multiplications?

If $G$ is a (possibly infinite) group, left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \rbrace \le \operatorname{Sym}(G)$ and $\Gamma:=\lbrace \gamma_a \...
1
vote
1answer
30 views

About monoid homomorphism

Let's say I have two sets $A$ and $B$, which are power sets of natural numbers less than or equal to $1$ and $2$ respectively. So $A$ = {$\emptyset$, {$0$}, {$1$}, {$0,1$}} and B = {$\emptyset$, {$0$},...
2
votes
1answer
38 views

Is a semigroup with unique right identity and left inverse a group?

We know that a semigroup with a right identity and right inverse for all elements is a group (e.g. see here). Symmetrically, also a left identity together with a left inverse implies a group. We also ...
1
vote
2answers
43 views

Showing ((a1 ⋄ a2) ⋄ a3) ⋄ a4 = a1 ⋄ (a2 ⋄ (a3 ⋄ a4)) in a semigroup.

I am a student in computer science - first year. I study linear linear algebra 2 - course of linear algebra 1. - In some institutions academic studies teach the courses together / teach in another way....
3
votes
1answer
68 views

Inverse of a product in semigroup

Disclaimer. The following might end up being a stupid question Let $S$ be a regular semigroup i.e for every $s\in S$ we can write $s=sas$ for some $a\in S$ (called an inverse of $s$). Do we know ...
4
votes
1answer
60 views

Commutative subtraction

It is well known that subtraction is not commutative in general. However, it is commutative in some groups: $\mathbb I$, $\mathbb C_2$, $\mathbb K_4$. I am trying to understand the logic. ...
2
votes
2answers
78 views

Show that $x^{mn}=x^{m}$

Let $(M,.)$ be an associative operation such that there are two natural numbers $m,n\in\mathbb{N}$, different from zero, such that $m$ is bigger or equal to $n$ and $x^{m}y^{n}=yx$, for any $x,y\in\...
0
votes
1answer
35 views

Let $G$ be a finite semigroup .Prove that there exist $x\in G $ such that $x^2=x$ [duplicate]

Let $G$ be a finite semigroup .Prove that there exist $x\in G $ such that $x^2=x$ How to approach this problem.i know i have to use that $G$ is finite set. but from where to start. please provide ...
1
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1answer
42 views

Is a semigroup with left identity und unique left inverse a group?

Let $(G, \cdot)$ be a semigroup (i.e. a set with an associative binary operation) and fix some $e\in G$. If 1) $\forall g\in G: e\cdot g=g$ (left identity), 2) $\forall g\in G~ \exists g^{-1}\in G: ...
4
votes
1answer
136 views

If there exists $m,n\geq2$ such that $x^{m}y^{n}=yx$, for any $x,y\in M$, then prove that the operation is commutative

Let $M$ be a set, not null, and $*$ an operation that is associative. If there exist $m,n\geq2$ such that $x^{m}y^{n}=yx$, for any $x,y\in M$, then prove that the operation is commutative. Now, I ...
5
votes
0answers
85 views

A counterexample to show that the following set does not form a semigroup

Let $A = $ { $f:\mathbb Z$ $\to \mathbb Z$| the cardinality of set {$x \in \mathbb Z$ | $f(x) = x$} is finite}. I have to prove or disprove that the set $A$ forms a semigroup/monoid under function ...
14
votes
0answers
127 views

Power of subset of finite group is a subgroup. [duplicate]

Let $G$ be a finite group and $S$ a nonempty subset of $G$. I want to prove (or disprove) that $S^{|G|}$ (that is products of length $|G|$ of elements of $S$) is a subgroup. My work so far : Since ...
0
votes
1answer
37 views

Product of two left invertible elements is also left invertible in Semigroup

Consider a Semigroup $(M, \ast)$ with a neutral element $e$. Now I have to prove that all left invertible elements of $(M, \ast)$ form a sub-semigroup. A left invertible element is an element whose ...
6
votes
0answers
80 views
+50

Bounding the number of certain (translation) subsets in $\mathbb N \times \mathbb N$ with respect to given subsets

Let $\mathbb N = \{0,1,2,\ldots\}$ be the monoid of natural numbers with zero. Suppose $S \subseteq \mathbb N \times \mathbb N$ be some subset such that the number of sets of the form $\{ (i,j) \mid (...
1
vote
1answer
43 views

Each ordered semigroup is cancellative: reference?

It is easy enough to show that $a+b < a+c\Rightarrow b < c$ holds in totally ordered semigroups. Indeed this must be very well known. Can anyone please provide a reference for this result? A ...
1
vote
1answer
28 views

Question on some specific property of ordered semigroup

Let $\langle S, \cdot, \leq \rangle$ be an ordered semigroup (or monoid). Suppose, we have some element $a \in S$, such that for each $b \in S$, $a \cdot b \leq a \cdot b \cdot a$ and $b \cdot a \leq ...
2
votes
0answers
71 views

Homomorphisms and semilattice decomposition of a band with an identity.

My question is fairly simple. Suppose we have a right-zero semigroup with an attached identity and decompose it into a semilattice. How do we define the structural homomorphism from 1 into the right-...
2
votes
1answer
45 views

Left Semigroups and Compact sets

I have fallen down the rabbit hole of reading old papers. I (believe that I) have read the following claim: Let $(S,\cdot)$ be a semigroup and assume that $S$ is also a topological space such that ...
1
vote
2answers
46 views

Is the inverse image of a group also a group for semigroup homomorphisms

If $\varphi : S \to T$ is a surjective semigroup homomorphism between semigroups and $G \subseteq T$ is a group, then is $\varphi^{-1}(G)$ also a group? I know that this result holds if $S$ and $T$ ...
0
votes
1answer
41 views

Right zero in a finite semigroup

Let $(M, \cdot)$ be a finite semigroup such that $$ x,y\in M\wedge \exists a,b\in M:x=a⋅y\wedge y=b⋅x\Rightarrow x=y. $$ Show that M contains at least one right absorbant element(or right zero). ...
2
votes
1answer
60 views

Is there a name for an associative algebraic structure in which everything is irreducible?

Let $A$ be a set and $\ast$ a binary operator on that set. Let us suppose that $(A,\ast)$ satisfies the following axioms: For all $x,y,z \in A$, $x \ast (y \ast z) = (x \ast y) \ast z$ For all $x,y,z ...
10
votes
1answer
127 views

Prove that a semigroup satisfying $a^pb^q=ba$ is commutative

Let $(S, \cdot)$ be a semigroup. There are natural numbers $p,q \geq 2$ such that $a^pb^q=ba$ for all $a,b \in S$. Prove that $S$ is commutative. I wrote $$\begin{align} a^{p+1}b^{q+1} &=b^{(q+...
1
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0answers
64 views

Herstein's Topics in Algebra Section 2.3 Problems 12 and 13

So, these are the questions: Let $G$ be a nonempty set closed under associative product, which in addition satisfies: a) There exists an $e \in G$ such that $a.e=a$ for all $a \in G.$ b) Give $a\in ...
1
vote
0answers
22 views

What is the non trivial example of monosemiring?

I considered the definition of monosemiring as: a semiring $(R, +, .)$ is said to be a monosemiring if $x+y=xy$ $\forall~x, y\in R,$ where $(R,+)$ and $(R, .)$ are semi groups. I also know that ...
0
votes
0answers
29 views

construction bijection and to understand Dyck path.

In here: https://oeis.org/A080936. I want to understand $T(n,k)$ is the number of Dyck paths of semilength $n$ and height $k$ and following triangle. \begin{align} 1;\\ 1, & 1;\\ 1, & 3, 1;\\ ...
2
votes
1answer
23 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
42 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
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0answers
21 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
38 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 ...
-1
votes
1answer
140 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
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0answers
39 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
29 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
112 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 ...
3
votes
1answer
72 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
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1answer
42 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
59 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
33 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
53 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
37 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
56 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
58 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
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1answer
48 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
126 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
49 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; ...