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.

learn more… | top users | synonyms (1)

1
vote
0answers
25 views

If $\lambda\subseteq\mathscr{L}, \rho\subseteq\mathscr{R}$, then $\lambda\circ\rho=\rho\circ\lambda$.

This is (the first part of) Exercise 2.4 of Howie's Fundamentals of Semigroup Theory. The Details. Quoting Howie (on page 22 of my copy): Definition: A relation $R\subseteq S\times S$ on a set ...
2
votes
0answers
96 views
+50

The congruence $\{(a^m, a^{m+r})\}^\#$ on $a^+$.

I've spent a bit too long on this exercise. It's time to ask for help. This is Exercise 1.20 of Howie's Fundamentals of Semigroup Theory. Let $\rho_{m, r}$ (for $m, r\ge 1$) be the congruence ...
4
votes
1answer
47 views

One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
-1
votes
1answer
25 views

Semigroup of a group and identiy

If $(G, *)$ is a group and $£$ is restriction of $*$ on subset $S$ of $G$. Is there some semi-group $(S, £)$ such that identity of $(S, £)$ and $(G, *)$ are different.
1
vote
1answer
19 views

Semigroup without left cancellation law and right cancellation law

Can please someone give example of a Semigroup in which neither left cancellation law hold nor right cancellation law
2
votes
2answers
20 views

Existence of right and left identity in minimalistic algebraic structure

Let $(A,\cdot)$ be some algebraic structure in which there exists elements $e_r,e_l$ such that $$e_l\cdot x = x, \forall x\in A$$ $$x\cdot e_r = x, \forall x\in A$$ By definition, if $(A,\cdot)$ is ...
-1
votes
1answer
85 views

Can a semigroup be extended to an arbitrary superset? [closed]

Let $S$ be a semigroup and $T$ be an arbitrary set with $S\subseteq T$. Is there a semigroup structure on $T$ with $S$ a subsemigroup of $T$?
2
votes
1answer
64 views

How would a category theorist describe Green's relations?

In Semigroup Theory, Green's relations are everywhere. Their equivalence classes, for instance, on a given semigroup $S$ can tell one a lot about the structure of $S$. There is some trivial sense in ...
0
votes
1answer
47 views

For semigroups, $S\preccurlyeq T$ iff there exists an injective relational morpism $\mu: S\to T$.

This is Exercise 1.16 of Howie's Fundamentals of Semigroup Theory. The Details. Definition 1: Let $A$ and $B$ be sets. A relation $\rho$ from $A$ to $B$ is a subset of $A\times B$. Define ...
1
vote
2answers
54 views

Involution on inverse semigroups

I'm trying to prove the following for inverse semigroups $\bf Def:$ an inverse semigroup $S$is a semigruop such that for each $x\in S$ the exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. An ...
1
vote
0answers
71 views

How many associative ternary operations there are on a finite set?

We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$ If $X$ is a set and and cardinality is $|X|=n$ then ...
2
votes
0answers
30 views

How to construct a semi-group over a non-numerical set?

A semigroup is a set, e.g. $X$, with an associative binary operation, e.g. $\star$. That is for all $x,y,z \in X$, $(x \star y) \star z = x \star (y \star z)$. I have seen some abstract algebraic ...
2
votes
1answer
65 views

About Rees homomorphism

I am came across the notion Rees Congruence for semigroups. J. Howie defines it as $$\rho_I=(I\times I)\cup {1_S}$$ wherein $I$ is an ideal of semigroup $S$ ...
2
votes
1answer
50 views

Showing the full transformation semigroup $\mathscr{T}_n=\langle\zeta, \tau, \pi\rangle$.

I'm sorry if this is a duplicate in any way. Throughout I use cycle notation and write maps $m:X\to Y$ on the right of their arguments (e.g. $xm=y$ for $m(x)=y$). This is Exercise 1.7 of Howie's ...
2
votes
1answer
38 views

Complicated proof in Transformation Semigroups.

Let $X$ be an infinite set. The relative rank of a subset $T_{X}$ over a subset $S$ is either uncountable or at most $2$. I don't understand how to prove this Corollary in case that $T_{X} \setminus ...
3
votes
0answers
58 views

Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...
2
votes
1answer
55 views

Commutative Cancellative Semigroup: When is an irreducible element prime?

Suppose $(S,\cdot)$ is a semigroup with neutral element $e$ (i.e. $xe=ex=x$ for all $x\in S$) and the following properties: S is commutative: $xy=yx$. S is cancellative: $xy = xz$ implies $y = z$. ...
1
vote
0answers
21 views

Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
1
vote
1answer
23 views

finite semigroup on one generator,cycle, tail,group,zero element

Suppose we have a finite semigroup on one generator. It has a tail of length r and cycle of length c.The cycle is a group, but what can be chosen as a neutral element of it?Why is not ANY element ...
0
votes
0answers
28 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
1
vote
1answer
34 views

How to prove that: If two binary operations are anti-isomorphic and one of them is associative then the second one also will be associative?

We know what is called an anti-isomorphic operation on a set S. it is just a one two one $ g $ function mapping from $S$ to $S$. $ g: S \rightarrow S$. and it satisfy this condition $ g(xy)= ...
0
votes
0answers
16 views

Haar measure on locally compact semigroups

I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure? ...
7
votes
3answers
166 views

Is there a name for a semigroup whose idempotents form a subsemigroup?

For a semigroup $S,$ I will denote by $E(S)$ the set of all idempotents of $S$. For $X\subseteq S,$ let $X^2$ mean $\{xy\,|\,x,y\in X\}.$ Is there a name for the class of semigroups $S$ such that ...
6
votes
1answer
118 views

About automorphisms of commutative semigroups

Suppose that $M$ is a commutative monoid and that the product $P$ of $M$ and the nonnegative integers $\mathbb{N}$ with addition has no nontrivial automorphisms. The set $S$ of pairs $(m,n)$ in $P$ ...
4
votes
1answer
113 views

Who first studied semilattices?

Historically, who first studied semilattices, as opposed to lattices or Boolean algebras? (With or without identity, I do not mind.)
2
votes
1answer
76 views

An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
3
votes
1answer
129 views

Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
3
votes
1answer
65 views

Period of semigroup

Let $S$ be a finite semigroup of order $n$. Suppose that $S$ has index $m$ and period $r$, i.e. $S$ satisfies the identity $x^{m+r} = x^m$. Then it is quite easy to show that $m \leq n$. My question ...
2
votes
3answers
49 views

Example of an inverse semigroup

An inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...
0
votes
1answer
25 views

Finite Linearly Ordered Abelian Monoids

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids". The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The ...
1
vote
1answer
29 views

Terminology for idempotents that commute with every other idempotent

Given a semigroup $S$, is there terminology for those $x \in S$ such that the following hold? $x$ is idempotent Given any idempotent $y \in S$, we have $xy=yx$. Comments. Let $E$ denote the set ...
2
votes
1answer
35 views

Help in this characterization of the gaps of the symmetric numerical semigroups

Before my question, some background: Definition 1: A numerical semigroup is a subsemigroup $N$ of the additive semigroup $\mathbb N$ of the non-negative integers such that $\mathbb N-N$ is ...
1
vote
2answers
30 views

Evolution Equation

Let $X=L^2(0,\pi)$. Define the operator $(A,D(A)$ by: $$D(A)=\{u\in H^2(0,\pi):u(0)=u'(\pi)=0\} ,\quad \quad Au=u''$$ Show that $A$ is the infinitesimal generator of a $C_0$ semigroup of contractions ...
1
vote
1answer
32 views

Example of a semigroup with unique idempotent which is not a monoid

I am searching for an example of a semigroup , with unique idempotent element , such that it is not a monoid . Please help
0
votes
0answers
16 views

Heat semigroup on Morrey spaces

I have a few questions concerning the heat semigroup $e^{\Delta t}$ on Morrey spaces. (1) I read that the heat semigroup on $ M^{p}_{q}(\mathbb{R}^n):= \left\{ f \in L^{q}_{loc}(\mathbb{R}^n): ...
0
votes
0answers
68 views

What are $\Gamma$-semigroups?

I have some problems with $\Gamma$-Semigroups, the definition that I've found is A $\Gamma$-Semigroup is a pair $(M,\Gamma)$ defined as follow If $x,y$ and $z$ are in $M$ and $\alpha$ and ...
2
votes
4answers
59 views

Showing uniqueness of inverse element of an element of a monoid

Question- If $\langle A,*\rangle$ is a semigroup with identity, prove that every element a belonging to $A$ has at most one inverse. Proof- Let the identity be $e$. Let us assume that $b_1$, $b_2$ ...
1
vote
0answers
28 views

Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in ...
5
votes
1answer
145 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
4
votes
1answer
189 views

A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
1
vote
0answers
37 views

Reference request on numeric semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
9
votes
1answer
266 views

Properties of Group $\to$ Monoid $\to$ Semigroup.

In How to get a group from a semigroup Arturo Madigin writes: So if we look at the composition of the forgetful functors $\bf Group \to \bf Monoid \to \bf Semigroup$, we obtain a right adjoint by ...
1
vote
1answer
39 views

Findout if $\min(x,y)$ and $\max(x,y)$ are semigroup, monoid or group

I need to find out whether $A \oplus B := \max(A,B)$ or $A \ominus B := \min(A,B)$ form a semigroup, monoid or group for $\oplus : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and $\ominus : ...
6
votes
1answer
83 views

About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
1
vote
1answer
65 views

If $G,H$ are monoids and not groups, prove that $f(e_G)=e_H$ may be wrong.

Problem: If $f:G\longrightarrow H$ is a homomorphism of groups, then $f(e_G)=e_H$ and $f(a^{-1})=f(a)^{-1}, \forall a\in G$. Show by example that the first conclusion may be false if $G, H$ are ...
8
votes
1answer
225 views

Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
3
votes
3answers
119 views

Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
2
votes
0answers
156 views

Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product A*B={PQ|P in A & Q in B & dim(Q)=codim(P)}, and ...
2
votes
1answer
22 views

Does a discrete semilattice of finite width have finite number of points between any pair of elements.

Let $(S,<)$ be a countably infinite semilattice such that: 1) $S$ is no-where dense - i.e. there does not exist a subset $T$ such that for all $a,b\in T$ with $a<b$, there exists $c\in T$ with ...
4
votes
2answers
148 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...