2
votes
0answers
93 views

Orbit-Stabilizer theorem for continuous groups

The orbit-stabilizer relationship (also known as the orbit-stabilizer theorem) is very clear for finite groups. Is there an equivalent relation for continuous groups? Also, is there a similar notion ...
2
votes
0answers
66 views

Semidirect products of semigroups [closed]

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function ...
4
votes
2answers
156 views

Embedding a linearly ordered free monoid into a linearly ordered group

A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...
3
votes
0answers
115 views

Computing the pro-solvable closure of a finitely generated subgroup of a free group

The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...
4
votes
1answer
133 views

Progress on group languages characterizations

Def. A group language is a recognizable language whose syntactic monoid is a group. q1. Is it known a "nice" combinatorial characterization of group languages ? q1.1. If no, is it well understood ...
24
votes
3answers
601 views

What is the probability two random maps on n symbols commute?

It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
2
votes
3answers
279 views

Generalized free product of semigroups with amalgamated subsemigroups

Hanna Neumann in [American Journal of Mathematics, 1948, http://www.jstor.org/discover/10.2307/2372201?uid=2&uid=4&sid=21102497379451 ] introduced a notion of generalized free product of ...
0
votes
1answer
170 views

Embedding a semigroup into a divisible semigroup

The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
2
votes
2answers
251 views

Projective limit construction of a semigroup

Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. ...
1
vote
0answers
85 views

Equivalence between Algebraic Semi-group Structures and Coalgebra Structures for an Algebraic Variety?

I was looking at this old question Hopf algebra and group structure correspondence for algebraic varieties which says that there exists an equivalence between algebraic group structures on an ...
0
votes
0answers
167 views
3
votes
2answers
210 views

Minimal right ideals in finite semigroup

Let $E$ be a finite semigroup. According to N. Bourbaki (Algèbre I p. 121 exerc. 14 c), if $M$ and $M'$ are minimal right ideals in $E$, then they are isomorphic. I spent some time browsing through ...
2
votes
1answer
259 views

On a property of subsemigroups

Let $H$ denote a subsemigroup of a semigroup $G$. I'm interested in the following property: $$\forall g\in G\exists h\in H:gh\in H.$$ This property is weaker than the property that $H$ is an ideal ...
19
votes
2answers
579 views

Mapping from a finite index subgroup onto the whole group

Dear All, here is the question: Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$? My guess ...
4
votes
0answers
212 views

What is the pro-algebraic completion of the free semigroup on one generator?

This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view. Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
7
votes
1answer
636 views

How is called a semigroup…

Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
3
votes
1answer
227 views

submonoid of a matrix monoid with a common eigenvector

Hello, I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ ...
0
votes
0answers
154 views

semigroup actions of groups on regular rooted trees

If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
5
votes
1answer
411 views

an example of a semigroup with solvable word problem but unsolvable power problem

We say that a semigroup $S$ has solvable power problem if there is an algorithm that takes as input an element $s \in S$ and decides whether or not there exist $m,n \in \mathbb{N}$ with $m \neq n$ and ...
14
votes
12answers
2k views

Why semigroups could be important?

There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
6
votes
2answers
280 views

need references regarding the elementary theory of free semigroup and free abelian groups

Recently, I read that two free abelian groups $S$ and $T$ have the same elementary theory if and only if rank$S$=rank$T$. Does anyone have a reference with a proof of this? Also, what is known about ...
0
votes
1answer
245 views

A Nomenclature Issue : Imprimitive Semigroup?

The following question was asked by me on the forum sci.math.research, “An imprimitive group is a transitive permutation group with a non-trivial equivalence relation compatible with the action of ...
2
votes
1answer
311 views

Residual finiteness of groups versus residual finiteness of semigroups

A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't ...
3
votes
2answers
295 views

What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes?

For a monoid $M$ and a subset $S$ of $M$, define the syntactic congruence $\equiv_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. : $$u \equiv_S v \Leftrightarrow (\forall x, y)[xuy ...
12
votes
1answer
1k views

Do these conditions on a semigroup define a group?

As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions ...