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 ...

learn more… | top users | synonyms (1)

3
votes
1answer
27 views

Semigroup where the sum of any two elements is one of the two elements

Is there a name for a semigroup where the sum of any two elements is one of the two elements? Let $S = (G, +)$ be a semigroup. I want to add the additional structure that $\forall g \in G, \forall h ...
1
vote
0answers
20 views

Bi-character for finite, commutative monoids?

If I have a finite commutative monoid $M$ (which is not a group), is it possible to get a bi-character on this? By bi-character, I mean a map $\beta:M\times M\rightarrow \mathbb{C}^*$ such that, ...
1
vote
0answers
11 views

Ring like structure with non-associative “addition”

Is there any formally defined algebraic structure which consists of a set and two binary operations, where one is a commutative magma ("addition"), and the other is a semigroup ("multiplication"), ...
2
votes
1answer
22 views

Need help understanding Proposition 2.3.7 from Howie's Semigroups.

I don't understand the author's argument in the second line of the proof. In particular, I don't see exactly how such a bijection does exist. I see that $ab \in L_b$ implies $H_b = H_{ab}$, and the ...
0
votes
0answers
40 views

Equality of two functions.

I have a specific question, from a paper given below. Here I got an answer of question: When two functions are called equivalent?.It helped me to understand the first and the second steps of the ...
1
vote
1answer
22 views

Strongly continuous group and generator commute, what about square roots?

Let $A$ be a positive self-adjoint operator, then $iA$ generates a unitary strongly continuous semigroup (Stone's theorem) $T$. Then from basic semigroup theory we know that $T$ and $A$ commute, but ...
1
vote
1answer
18 views

Extending Semiring $\mathbb{N}$ to $\mathbb{Z}$ through exact sequence

I am working on extensions in the form of $$A\hookrightarrow B\twoheadrightarrow C$$ in my thesis and I am just wanting to add as an extra note, IF POSSIBLE, this. We have that that $\mathbb{Z}$ is ...
-2
votes
1answer
33 views

Prove that a semigroup which satisfies a certain conditions is a group [closed]

This is an exercise from "Abstract Algebra" by P.A.Grillet (p.12, ex.2). Let $S$ be a semigroup (that is, a set with an associative binary operation) in which there is a left identity element ...
0
votes
0answers
32 views

Regularity of the solutions of the infinite dimensional dynamical systems

Consider a densely defined unbounded operator $A:D(A)(\subset H)\to H$ which is infinitesimal generator of a strongly continuous semigroup $\mathbb{T_{t\ge0}}$ for the following dynamical system: ...
4
votes
2answers
68 views

What's the order of a semigroup?

For a group the order, of an element is the smallest positive integer m such that a^m = e. But what's the order of an element of a semigroup? Or there isn't anything like that?
8
votes
1answer
89 views

The lattice points in the real cone of some semigroups are just the integer cone of that semigroup.

I'm trying to solve an exercise in Fulton's book on toric varieties, and have reduced it to the following: Let $M$ be a lattice of rank $n$ with $M \otimes \mathbb{R} = V$, and $S$ be a finitely ...
0
votes
3answers
87 views

What is the one-one and onto mapping?

I'm reading the following paper, and here I can't understand the first step of the algorithm. Please explain what is the one-one and onto mapping? And how he gets these tables ? Commutative ...
1
vote
0answers
20 views

Looking for concrete examples of semigroups, with explicit computations done to help make the theory more understandable.

I would appreciate if someone can help me find a resource which has a bunch of worked out examples showing how to find, for example, the smallest congruence containing a relation on the semi groups, ...
1
vote
1answer
76 views

Is it possible to say that if a A is a set of elements of a semigroup S, there exists some other semigroups on the alphabet A?

Suppose we have a finite semigroup, its element's set is A = {1,2,...n}, can we get some other semigroups whose element's set is A too ?
1
vote
1answer
33 views

A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
0
votes
1answer
90 views

Is there any efficient algorithm for computing all semigroups of order n?

I found a paper, but here the author solves a bit different problem. My question is: Is there any efficient algorithm for computing all semigroups of order n?
0
votes
1answer
58 views

A group specified by Generators and Relations.

I'm confused with some terms in several definitions. Is an alphabet of a free group the same thing as a generator set of any group? If it is right, then by a given alphabet (set of generators) can be ...
2
votes
1answer
28 views

The code SetReducedMultiplication for semigroups

Let $S$ be a finite semigroup like this: ...
2
votes
2answers
76 views

Proving a semigroup to be abelian.

If in a semigroup S,$ \ $ $ x^{k+1} = x $ for some $k \ge 1$ and $xy^kx = yx^ky \quad \forall x,y \in S$ then show that S is abelian. I'm able to prove the following $\ x=x^3, \quad ...
0
votes
0answers
24 views

Clifford semigroups- again!

Can someone help me regarding this question: If $S$ is a Clifford semigroup then $S=\cup S_{e_{i}}$, $e_{i} \in E(S), i=1,2,...n$, , $E(S)$ the set of idempotents of $S$. From a theorem in Howie its ...
4
votes
1answer
75 views

GAP tells this semigroup is not a group.

Happy Nowruz 2016 to every one here! Using the code which James pointed here; I was playing with the following finite semigroup: ...
1
vote
0answers
16 views

Realizing $\mathbf{C}$ as 1-dimensional normal toric variety

Take $N = \mathbf{Z}$, $N_\mathbf{R} = \mathbf{R}$. Then I know that the only cones are the intervals $\sigma_+ = [0,\infty)$, $\sigma_- =(-\infty,0]$, and $\tau = \{0\}$. Consider the fans ...
0
votes
0answers
17 views

Semigroup of a common face of two cones in a fan

Let $\Sigma$ be a fan, $\sigma_1, \sigma_2 \in \Sigma$, and $\tau = \sigma_1 \cap \sigma_2$ be a common face of the two cones. For a cone $\sigma$, denote the semigroup by $S_\sigma = \sigma ^\vee ...
3
votes
1answer
67 views

Do we have accepted terms for semigroups and semigroupoids without identities?

I would like to call a semigroup without an identity a "pure semigroup" and a semigroupoid without any identity a "pure semigroupoid". I'm not sure whether such things have been defined in literature ...
1
vote
1answer
18 views

Greatest Common Divisor Semigroups

I'm trying to find a proof that shows if $a$, $b$ are in the natural numbers, then the sum of the additive semigroups $\mathbb N a + \mathbb N b$ is a subset of $\mathbb N d$ where $d = \gcd(a,b)$.
4
votes
2answers
75 views

Every inverse semigroup is a group

The Wikipedia page about inverse semigroups defines them as follows: In mathematics, an inverse semigroup (occasionally called an inversion semigroup) $S$ is a semigroup in which every element ...
3
votes
1answer
40 views

Example of a monoid having at least 2 members for which $xy = 1$ but $yx \not= 1$

Can a monoid have at least 2 members for which $xy = 1$ but $yx \neq 1$ ? I tried matrix multiplication but $ AX = I$ then $XA = I $ too.
1
vote
1answer
34 views

Nilpotent Elements In Partial Maps Semigroup

Let $X$ be a set. The full transformation semigroup $T_X$ is the set of all maps from $X$ into $X$ with composition of maps as binary operation. A map $\alpha \in T_X$, is a subset of the cartesian ...
4
votes
2answers
42 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...
0
votes
1answer
19 views

free semi-group

I'm trying to prove the following : Let S be the free semi-group on the alphabet $ A$ and let T be an arbitrary semi-group. Assume that $ g : A \rightarrow\ T $ is any mapping. Prove that there is ...
0
votes
0answers
34 views

Semi lattice of groups!

Can someone please explain to me: How can I connect the structure of the Clifford semigroup with the structure of its maximal group image? Thanks!
3
votes
1answer
35 views

Left vs right projective resolutions and homology of monoids

Let me use the ad hoc notation $\mathbb Z^l$ and $\mathbb Z^r$ to distinguish between left and right modules. These are trivial modules. The homology of a (discrete) finite monoid $M$ with ...
1
vote
1answer
28 views

Prove that $\sigma^*$ is the least group congruence on $S$

Let $S$ be an inverse semigroup and consider the relation $\sigma$ on $S$ given by $$a \sigma b \iff ab^{-1} \in E(S)$$ Consider the congruence generated by $\sigma$, say $\sigma^*$. Prove that ...
2
votes
1answer
34 views

Clifford semigroups!

Can someone come up with a drawing how can Clifford semigroup be presented as a decomposition of $S_{\alpha_{i}}$, where $\alpha_{i} \in E(S)$, where $E(S)$ is the set of idempotents? I've gone ...
0
votes
1answer
35 views

Semigroups on Complex Hilbert Spaces

Let $H$ be a separable complex Hilbert space, let $(e_i)_{i\in\mathbb{N}}$ be a complex othonormal vasis, and let $(\lambda_i)_{i\in\mathbb{N}}$ be a sequence of complex numbers s.t. $\sup_{i\in ...
2
votes
1answer
63 views

Representation of regular languages by monoids [closed]

I'm interested in representation of regular languages by monoids, and in particular of how to use this kind of representation to get a recognizer. I have found some references on the web, but does ...
1
vote
1answer
27 views

Regarding Compact Semigroup

I have a problem in understanding the proof of following Theorem 2 (b), which is given in the book "The Theory of Topological Semigroups" by J. H. Carruth, J. A. Hildebrant and R. J. Koch as Theorem ...
0
votes
0answers
30 views

Multiplicative closure of convex set of matrices

Given a convex polytope of matrices $C$ is there a way to find its multiplicative closure $C^{*}$ such that $C^{*}$ is the smallest set which contains all products of sequences of matrices from $C$. ...
0
votes
0answers
22 views

Left Reversible Semigroups

A semigroup $S$ is left reversible if $aS \cap bS \not= \emptyset$ for every $a$, $b$ in $S$. Let $S$ be a nonempty left reversible semigroup and $S_0 \subseteq S$ be a proper subsemigroup of $S$. ...
0
votes
0answers
14 views

What are some applications of transformation semigroups?

I have been told that transformation semigroups have applications to statistics, computer science, and combinatorics. What are some basic (and if possible, simple) examples of transformation ...
0
votes
0answers
46 views

Subsemigroup of permutable semigroup!

Let $S$ be a semigroup and $n$ integer greater than 1. A sequence $s_{1}, s_{2},...,s_{n}$ of $n$ elements of $S$ is called permutable if the product $s_{1}s_{2}...s_{n}$, remains invariant, under ...
1
vote
2answers
32 views

Nonminimal idempotents

Let $S$ be a semigroup with operation $+$. Note that we do not assume $S$ is abelian. We say that $I\subset S$ is a left ideal if and only if for all $s\in S$, $s+I\subseteq I$, and we say that an ...
8
votes
2answers
270 views

What does it mean when people say that groups are a study of symmetry?

I see many people make remarks to the effect that groups have basic symmetry properties. I am familiar with Cayley's Theorem and the symmetric groups $S_N$. However, when I think of the symmetric ...
2
votes
1answer
51 views

Finitely generated Clifford semigroup

If $S$ is a finitely generated Clifford semigroup and $S'$ a subsemigroups of $S$, how can I prove that $S'$ is also finitely generated?
1
vote
2answers
41 views

In a commutative semigroup, $a_{\phi(1)}a_{\phi(2)}…a_{\phi(n)}=a_{1}a_{2}…a_{n}$

In the book of Clifford and Preston,"The algebraic theory of semigroups" volume I, I am reading this: If $a_{1},a_{2},...,a_{n}$ are elements of a commutative semigroup $S$ and $\phi$ is any ...
0
votes
0answers
40 views

Isomorphism in Clifford semigroups

If I take a subsemilattice $S'$ of the Clifford semigroup $S$, such that the groups of $S'$ are abelian by finite of finite index in the respective groups of $S$ and I want to prove that $S$ is ...
0
votes
0answers
39 views

S an inverse semigroup with semilattice of idempotents E and $\sigma$ the minimum group congruence on S.

Let S be an inverse semigroup with semilattice of idempotents E, and let $\sigma$ be the minimum group congruence on S. Show that the following statements are equivalent: (a) $x\sigma y$; (b) ...
4
votes
1answer
30 views

D-􀀀classes in an inverse semigroup are ‘square’.

Show that the $\mathcal{D}$-classes in an inverse semigroup are ‘square’. More precisely, show that there is a bijection from the set of $\mathcal{L}$-classes in a $\mathcal{D}$-class $D$ onto the set ...
1
vote
1answer
40 views

Is a subsemigroup of a Clifford semigroup also a Clifford semigroup?

Is a subsemigroup of a Clifford semigroup also a Clifford semigroup? I can prove that the maximal group image of a Clifford semigroup is a Clifford semigroup, but I am not sure whether any ...
1
vote
0answers
51 views

Semigroup of matrices and expander Cayley graphs

I am interested in proving or disproving that certain Cayley graphs are expander. Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & ...