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.

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Strongly continuous semigroup on $(0,\infty)$ which is not continuous at $t=0$.

In most cases a strongly continuous semigroup of positive linear contractions on $L_1(X,\mu)$ $T_t$, $t\gt0$ on $(0,\infty)$ can be completed to a strongly continuous semigroup $T_t$, $t\ge0$ on $[0,\...
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12 views

partial injective transformation [closed]

Let $X_n=\{1,\ldots,n\}$. All of the partial injective transformation on $X_3$ at most $1$ image as following: $ \begin{pmatrix} 1 & 2 & 3 \\ 1 & - &- \end{pmatrix},\begin{pmatrix} ...
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1answer
81 views

Is $*$ associative?

Suppose $*$ is a binary operator on a set $A$ such that $\forall x,y\in A,$ we have $$x*(x*y)=y$$ and $$(y*x)*x=y.$$ Is $*$ associative? I can show that $*$ is commutative, because each of the ...
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35 views

What is the correct name for a “product function” on a monoid?

Let $W$ be a monoid. A function $f\colon W\rightarrow W$ is a "product function" if $f(w)$ is a product of constants in $W$ and positive integer powers of $w$. It could also be called a "non-...
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1answer
59 views

Ring homomorphism may not preserve $1$.

Suppose $R$ is a ring with unity,and $f:R\to R'$ is a ring homomorphism.Then $f(R)$ must have a $1_{f(R)}$ .But $1_{f(R)}$ may not be the identity of $R'$.Even $R'$ may not contain any identity and ...
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1answer
17 views

Show there exists a non-commutative cancellative semigroup with generators $w,x$ satisfying $xwx=ww$

Let $W$ be a cancellative semigroup and $w,x\in W$ non-identity elements such that $xwx=ww$ and $\{x,w\}$ generates $W$. Prove that $W$ is commutative or find a counterexample. (Note that $x=1,w=2,W=\...
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1answer
180 views

Show that $H_x$ is a group for all $x.$

$H_{x}$ denotes the class of $x$ for the Green relation $\mathcal{H}$. Let $S$ be a finite semigroup where all elements can be written as a product of idempotents, that is, $x=e_1 e_2\dots e_n,$ ...
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1answer
56 views

Show that $\mathbf{D}(\mathbb{C}\bigoplus \mathbb{C})$ is isomorphic to the additive semigroup $\mathbb{Z}^+\bigoplus \mathbb{Z}^+.$

I am reading "An introduction to $C^*$ Algebra" by Rordam. Show that $\mathbf{D}(\mathbb{C}\bigoplus \mathbb{C})$ is isomorphic to the additive semigroup $\mathbb{Z}^+\bigoplus \mathbb{Z}^+.$ I don'...
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59 views

Commuting fraction of $F_n$

Suppose $S$ is a finite semigroup. Let's define the commuting fraction $cf(S) := \frac{|\{(a, b) \in S^2| ab = ba\}|}{|S|^2}$. Alternatively, it can be defined as the probability that two independent ...
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1answer
35 views

Surjectivity of right and left translations implies semigroup is in fact group

I want to show a set of statements on a semigroup $G$ are equivalent. The left and right translations are given by $l_g(h)=gh$ and $r_g(h)=hg$ respectively. $G$ is a group For all $g \in G$ both $l_g$...
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2answers
49 views

Bounded strictly totally ordered semigroup

Is it possible that a strictly totally ordered ($<$) infinite algebraic structure has both maximum and minimum? There is an example of a strictly totally ordered infinite magma bounded from two ...
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1answer
43 views

Elements of a semigroup/ring that are powers of each other

There is a special term for elements of a ring that are multiples of each other: "associates". In a wider context associates are elements of a semigroup that generate the same ideal. The equivalence ...
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59 views

Construct well-defined bijection

Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($...
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24 views

Factorization of the identity element of the free monoid

Let $X$ be a set and let $\text{Mo}(X)=\bigcup_{n\in\mathbb{N}} X^{[1,n]}$. Then $\text{Mo}(X)$ together with the law $(w,w')\mapsto ww'$, where $ww'$ denotes the juxtaposition of the sequences $w$ ...
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1answer
77 views

Given the following two properties of the multiplication table, show that $G$ is a group.

I came across the following problem: Note the $1$ below is defined to be the unit element. That is, $1\cdot g=g\cdot 1=g$ for all $g\in G$. Let $G$ be a finite set with a binary composition ...
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1answer
32 views

Finitely generated subsemigroups of $\mathbb{N}^k$

It is well-know that every subsemigroup of $(\mathbb{N},+)$ is finitely generated. I am wondering if there is (any) similar characterization of subsemigroups of $\mathbb{N}^k$ for $k>1$? I am ...
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12 views

On the number of connected functional digraphs with the same iterated preimage structure

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, $P_j=\left[f^{-j}(...
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2answers
60 views

The axiom quotient and any property of a group.

Def. A group $G$ is a set not empty in which is definited a binary operation $G\times G\to G$ denoted with $(a,b)\mapsto ab$ such that $$(i)\quad(ab)c=a(bc)\;\text{for all}\; a,b\in G;$$ $$(ii)\quad\...
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27 views

Semigroup generated by elements $x_1,…,x_n$ with $\mathrm{gcd}(x_1,…,x_n)=1$

Let $x_1,...,x_n\in\mathbb{N}$ be such that $\mathrm{gcd}(x_1,...,x_n)=1$ and suppose that $S$ is the semigroup generated by them. I would like to show that there always exists $x\in S$ such that ...
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2answers
48 views

example of monoids

An element $x$ of a semigroup $S$ is called regular provided that there exists $y\in S$ such that $xyx=x$. $S$ is called regular if all its elements are regular. Let $S$ be a monoid with identity ...
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2answers
100 views

How to show convolution is associative?

Consider a semigroup $\Gamma$ and the space $$l^1(\Gamma) := \left\{f: \Gamma \to \mathbb{C}: \sum_{x \in \Gamma} |f(x)| < \infty\right\}$$ where the summation is understood as in the following ...
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22 views

Is ultrafilter convolution a special case of convolution of probability measures?

If $G$ is a semigroup and $\beta G$ is the set of ultrafilters on $G$, then $\beta G$ is also a semigroup: given $p,q\in \beta G$, we define $$p*q := \{E\subseteq G : \{g\in G : g^{-1}A\in q\}\in p\}.$...
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32 views

One-sided monoids and groups [duplicate]

It is known that a semigroup with left identity element and left invertibility is a group. I noticed that I can make an unnamed group-like structure with some tweaks. I call a semigroup $\langle G,*\...
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73 views

Can you collapse commutativity of pseudo inverses with associativity?

I've been studying the possibility of collapsing axioms and I'm stuck trying to prove something (by collapsing two axioms i mean find an axiom equivalent to them). Take the following characterization ...
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0answers
35 views

prove that given map is isomorphism

Let $X$ be a set and $T_X$ be the full transformation semigroup on X. If $e\in T_X$ an idempotent, and if $Y=\text{image}(e)$, then the map $eT_Xe\rightarrow T_Y$ $f\to f|_{Y}$is an ...
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1answer
38 views

Ring graded by a non-Abelian monoid

I'm looking for interesting examples of a $G$-graded ring where $G$ is a non-Abelian semigroup, monoid or group. Obvious examples are the semigroup algebra $kG$, but I haven't come across any others. ...
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1answer
45 views

quasi ideal in semigroup

I am new in semigroup theory and I have an problem related to quasi ideals. I am trying to solve the following problem Problem: For a semigroup $S$, the following are equivalent $S$ is ...
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2answers
26 views

Finite semi-group generating set

Let $S$ be a finite semi-group which basically means that it is an associative binary operation. $A$ will be called as a generating set of $S$ if every element can be written as a multiplication of ...
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2answers
54 views

Prove that every abelian cancellation semigroup can be imbedded in a group

For an abelian cancellation semigroup $G$ and a group $F$, I think it is sufficient to prove that a homomorphism $f: G \to F$ must be injective. Following that line of reasoning, I want to say ...
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4answers
78 views

Idempotent, commutative and invertible

Is there a mathematical class and structure in which there exist many objects that are distinct, invertible, commutative and idempotent? Like a set of toggle switches with no hysteresis, so the state ...
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0answers
23 views

Congruence on semigroup [duplicate]

For a commutative semigroup $S$, define the relation $\theta_n^S\ \ (n\geqslant 1)$ by $$a\theta_n^Sb\text{ if and only if } (\forall x\in S^n) xa = xb. $$ (a) Show that $\theta_n^S$ is a congruence ...
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72 views

left and right Ore

A monoid $M$ satisfies the left Ore condition if for all $a,b \in M$ there exist $c,d \in M$ such that $ca=db$. Suppose, in addition, $ac=bc$ implies that there exists an element $d \in M$ such that $...
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1answer
55 views

On finding a finite set of generators for a certain semigroup

Let $A$ be a finite subset of $\mathbb Z^2$. Let $\mathbb ZA$ be the subgroup of $\mathbb Z^2$ generated by $A$. Let $\mathbb R_{+}, \mathbb Q_{+}$ denote the set of non-negative real and rational ...
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2answers
40 views

What is the definition of generating the same ideals?

Taken from An Introduction to Semigroup Theory by J.M. Howie, aLb if and only if a and b are generating the same principal ideals. What does it really means to generate the same ideal? If someone ...
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1answer
41 views

When are a set and its complement both syndetic?

Let $G$ be a semigroup. A subset $S\subseteq G$ is syndetic if $G$ is covered by finitely many translates of $S$: i.e. there are elements $g_1,\ldots,g_m\in G$ such that $G=Sg_1\cup \cdots\cup Sg_m$. ...
2
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2answers
90 views

Let $S$ be a semigroup. If every finitely generated $T\lt S$ is embeddable in a group then $S$ is embeddable in a group.

Let $S$ be a semigroup. If any finitely generated $T\lt S$ is embeddable in some group $G_T$ then also $S$ is embeddable in some group $G$. I am trying to prove this statement, which is an exercise ...
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1answer
55 views

Is the semigroup of lines $\mathcal{M}_n(\mathbb{Q})$, finitely generated

I was researching $\mathcal{M}_n(\mathbb{Q})$, the set of square $n\times n$ matrices with rational entries, as a semigroup with matrix multiplication. For $A,B\in\mathcal{M}_n(\mathbb{Q})$, the ...
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1answer
41 views

Group of mappings

Is there a group $G$ of mappings $X \to X$ that has a non-bijective map in it? I mean, for each element of G, it must has its inverses at right and left, and those must be the same, so the element is ...
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2answers
54 views

Existence of a commutative inverse semigroup with no identity element

Does there exist a commutative inverse semigroup with no identity element, or are all commutative inverse semigroups abelian groups? If there does, what would be an example of such a commutative ...
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1answer
69 views

A functional equation on a semigroup

Let $(\mathbb{R^{*}},\circ)$ be a semigroup having the property that $\forall x\in \mathbb{R^{*}}, \exists y \in \mathbb{R^{*}}$ such that $x \circ y \neq y \circ x$. Find the functions $f: \mathbb{R^{...
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1answer
61 views

Are positional notation systems for natural numbers wreath products of semigroups?

Suppose we are given the finite cyclic group $\mathbb{Z}/b\mathbb{Z}$ and the monoid of natural numbers $\mathbb{N}$, both of which are semigroups. Does the restricted wreath product $(\mathbb{Z}/b\...
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2answers
62 views

The monoid of fractions associated with the submonoid of cancellable elements of a commutative monoid E

Let $E$ be a commutative monoid, $\Sigma$ the submonoid of cancellable elements of $E$, $E_{\Sigma}$ the monoid of fractions of $E$ associated with $\Sigma$ and $\varepsilon$ the canonical ...
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1answer
43 views

Proof that commutative groupoid $(G,*)$ is a semigroup

Prove that the commutative groupoid $(G,*)$ is a semigroup if the following equation is true for every $x,y,z \in G$ : $$(x*y)*z=(z*x)*y$$ So what I know is that for a groupoid to be a semigroup we ...
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2answers
60 views

Is $\mathbb{R}\setminus \{-1\}$ a Semi group/Monoid/Group with $a\cdot b=a+b+ab$?

Is $\mathbb{R}\setminus \{-1\}$ a Semi group/Monoid/Group Where $a\cdot b=a+b+ab$? So we have to show: 1. that the operation is closed and associativity, then to find an identity element, and then if ...
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1answer
11 views

Large semigroups of real matrices with real eigenvalues

There are a bunch of matrix properties that ensure that all the eigenvalues are real, chief among them being symmetric. However, I cannot find nontrivial examples of semigroups (under any kind of ...
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0answers
22 views

Semilattice of idempotent

Let $E$ be a well ordered chain semilattice of idempotent ( $E=\{e_{0}, e_{1}, ... , e_{n}, ... \}$ where $e_{i}\leq e_{j}$ if $i\leq j$). Prove or disprove if $e_{k}e_{h}e_{m}=e_{h}e_{m}$ and $ e_{k}...
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1answer
40 views

Associative binary operation with several left inverse elememts

I am supposed to find a binary operation that is associative and has several left inverse elements. I have no clue how to do that or if it is even possible, please help :)
3
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1answer
63 views

Example of a monoid required for union operation

Consider $(G,\cup)$ to be a groupoid defined with respect to the union operation. Let $G=\{A,B,C\}$ , where $C= A \cup B$ and $A$, $B$ are any arbitrary sets. $(G,\cup)$ is closed. Associative law ...
3
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2answers
177 views

Semigroup over a finite set that is nothing more than that

Ok so a semigroup only has associativity. Suppose i have a set $S = \{a,b,c,d\}$ Someone give me an example for the function $+:S \times S \rightarrow S$ (in table form) so that is only associative ...
2
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7answers
87 views

Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these?

Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these? I tried to solve it by assuming $ a,b,c \in G $ such that $a*(b*c)=(a*b)*c$. Then$$\frac{a+\frac{b+c}{2}}{2} = \frac{...

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