0
votes
1answer
18 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 ...
6
votes
1answer
81 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 ...
6
votes
1answer
192 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 ...
-1
votes
0answers
20 views

Quotient splits in direct product

I have a quotient, say $$Pert(\mathcal{A} \oplus \mathcal{B}) / \ker(\phi) \cong Im(\phi) = Pert(\mathcal{A}) \times Pert(\mathcal{B}),$$ and I know that $$Pert(\mathcal{A} \oplus \mathcal{B}) \cong ...
4
votes
2answers
140 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 ...
2
votes
1answer
62 views

under what conditions a product of matrices is the identity matrix (more complicated than that)?

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple ...
3
votes
0answers
50 views

semidirect product of semigroups

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 ...
1
vote
1answer
69 views

Is Writing a Semi Group?

While writing this question, I compose letters from a set $L=\{a,...z,A,...Z\}\cup\{\;\text{ } \;\}$. Writing has a binary operation which is associative. The result always is an element of $L^n$. ...
0
votes
1answer
26 views

Proof of no identity element in a multiplication semi group

I have a question from my workbook: Let $\mathrm{E}$ denote the set of even integers. This forms a semi-group under multiplication. Show that there is no identity in this semi-group. Now this is ...
0
votes
1answer
45 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
0
votes
0answers
14 views

What is the semigroup reduct of an abelian torsion group

I was going through a paper on universal algebra where the author mentions one of his examples as a the "semi group reduct of an abelian torsion group". I have no idea what a semigroup reduct means ...
1
vote
1answer
32 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 : ...
2
votes
0answers
26 views

How to calculate when a specific semigroup is closed under conjugation by this group.

Let $\mathcal{T}_n$ be the set of $n \times n$ matrices such that $X\in \mathcal{T}_n$ can be written as: $$X = \bigl( \begin{array}{cc} 1 & \overline{0} \\ T & \underline{0} \end{array} ...
4
votes
1answer
102 views

Naturally Ordered Semigroup: Why does axioms imply order of group is infinite countable ? Why are every group equal up to isomorphism?

Naturally Ordered Semigroup: http://www.proofwiki.org/wiki/Definition:Naturally_Ordered_Semigroup I know $\mathbb N$ is a naturally ordered semigroup by definition. Also I know that the naturally ...
2
votes
1answer
52 views

Necessary and sufficient conditions for the embeddability of a semigroup in a group

According to wikipedia, The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in (Malcev 1939).[5] Though theoretically important, the ...
2
votes
1answer
70 views

If $S$ is a right group, then every $\mathcal L$-class is a $\mathcal H$-class and also a subgroup.

A semigroup $S$ is called right simple if it contains no proper right ideal. A semigroup that is right simple and left cancellative is called a right group. This is equivalent to saying that, for any ...
4
votes
1answer
96 views

definition of (semi)group (co)homology

I'm puzzled why "group cohomology" contains terms 'group' (instead of 'semigroup') and 'cohomology' (instead oh 'homology and cohomology'). I'm new to the subject. Please inform me of any claims ...
3
votes
1answer
203 views

G is a group if and only if for all $a,b∈G$, $ax=b$ has solution; true or false? Why?

I think we only need $ax=b$ or $ya=b$ have solutions in $G$, I'll prove it. Proof: (I) $G$ is a group $\implies$ $a^{-1}ax=a^{-1}b$ $\implies$ $x=a^{-1}b$ $\implies$ $ax=b$ has solutions in $G$ ...
3
votes
3answers
108 views

How could I understand this sentence:

How could I understand this sentence: For any subgroup $H$ of a group $G$ we have $H^2=HH=H$. However, in general, the same is not true for subsemigroups of semigroups. Thanks a lot!
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 ...
7
votes
3answers
420 views

Is a semigroup $G$ with left identity and right inverses a group?

Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group? Now, If both the identity and the ...
6
votes
1answer
114 views

Structure theorem for finitely generated commutative $semi$groups.

$\newcommand{\ZZ}{\mathbb{Z}}$ It is a classical theorem that a finitely generated commutative group is isomorphic to one of the form: $$ \ZZ^n \oplus \ZZ/{m_1}\ZZ \oplus \ZZ/{m_2}\ZZ \oplus \dots ...
4
votes
1answer
227 views

Structure Descriptions (GAP) in semigroups

As can be easily seen through GAP manual: 37.6 Structure Descriptions StructureDescription( G ) A The method for StructureDescription exhibits the ...
1
vote
4answers
201 views

Inverse elements in the absence of identities/associativity.

Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows. $(xy)z=x(yz).$ $xe=ex=x.$ ...
2
votes
1answer
189 views

Rees matrix semigroup

Let $\mathcal{M}^0 = \mathcal{M}^0 ( G; I , \Lambda ;P)$ be a Rees matrix semigroup ($G$ a group, $I$, $\Lambda$ non-empty sets, $P=(p_{\lambda i})$ a $\Lambda \times I$ matrix over $G \cup \{0\}$ ...
9
votes
1answer
259 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
58 views

Definition of tautological action

What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions? For reference the particular sentence is: "A ...
13
votes
5answers
1k views

Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...
0
votes
0answers
77 views

Is there a name for a subset $S$ of a group or a semigroup such that every two elements of $S$ commute?

Let $G$ be a group and $S$ its subset. I would like to consider the following condition on $S$. For every $x,y\in S,$ we have $xy=yx.$ This is trivially equivalent to $S\subseteq C(S).$ The ...
1
vote
2answers
213 views

If every element of a monoid is right invertible, then every element is invertible

Let $G$ be a set with associative binary operation and a unit. Assume that for every $g\in G$ there exists $x \in G$ with $gx = 1$. Prove that $xg = 1$ is a consequence.
3
votes
2answers
998 views

If cancellation laws hold, then a finite semi-group is a group [duplicate]

Show that if both cancellation laws i.e $w.a = w.b \implies a = b$ and $a.w = b.w \implies a = b$ holds then a finite semi-group (a finite set with associative binary operation) is a group. I have ...
1
vote
1answer
318 views

Why isn't there much study on right groups (left groups)?

This question is related to and inspired by the question Why are groups more important than semigroups?. I am curious why I don't see much studies on right groups. On Pp.37 of Clifford& ...
7
votes
4answers
1k views

Group identities and inverses

A set is a set. A magma is a set with a binary operator. A semigroup is a magma with an associative binary operator. A monoid has a two-sided identity. And a group has two-sided inverses. I am ...
8
votes
3answers
263 views

give a counterexample of monoid

$G$ is a monoid,$e$ is its identity,if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$. if not,please prove $b=c$ thanks a lot.
2
votes
4answers
598 views

Cayley Graph of Semigroups

I did Google search and can't find a good answer. I thought I should ask experts here. http://en.wikipedia.org/wiki/Cayley_graph is for groups. My question is, Is there a special name for the ...
11
votes
0answers
627 views

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

I am studying Scott's book Group Theory. In the Exercise $1.1.17$ he asks us to show that if $S$ is a set and $|S|=n$, then there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$. But ...
12
votes
1answer
312 views

clarification on the definition of meaningful product

I am studying Hungerford's book "Algebra". In the page 27 he defines the meaningful product as follows. Given any sequence of elements of a semigroup $G, > \{a_{1},a_{2},\dots\}$ define ...
49
votes
6answers
2k views

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
26
votes
3answers
935 views

What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
7
votes
1answer
244 views

When a semigroup can be embedded into a group

Under what assumptions can a semigroup $(S,*)$ be embedded into a group?
17
votes
2answers
598 views

How to get a group from a semigroup

I am sorry if my question is too simple. Is every semigroup associated to a group? If no, what conditions should be satisfied for a semigroup to have an associated group? If yes, how can I find the ...
5
votes
0answers
247 views

Inverse of the Grothendieck group construction?

Is there an "inverse" of the Grothendieck group construction that would generate a (somehow "simplest") Abelian semigroup given some (suitably qualified, if necessary) Abelian group? (I realize that ...
14
votes
2answers
1k views

(Organic) Chemistry for Mathematicians

Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do ...
5
votes
1answer
165 views

is there an analogue of short exact sequences for semigroups?

Since semigroups don't need to have an identity element, I was wondering if there's any kind of short exact sequence for semigroups.
4
votes
1answer
163 views

Semigroup with “transitive” operation is a group?

I have a semigroup $G$ (a set with associative binary operation) such that for all $a,b\in G$ there exists $x,y\in G$ such that $ax=ya=b$. Is this property enough to show that $G$ is a group, and if ...