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 ...

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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 ...
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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, ...
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10 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"), ...
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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 ...
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1answer
20 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 ...
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How to Calculate Flat rate using Effective Rate? [closed]

Following calculations are made thriugh an Excel calculator in my company. I had to input EMI to calculate Flat Rate , but I want asolution to calculate Flat rate straight from Full/Effective rate ...
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1answer
17 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 ...
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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: ...
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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 ...
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1answer
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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 ...
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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?
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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 ...
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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 ...
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1answer
73 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 ?
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85 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?
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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, ...
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26 views

Difference between the set of generators and the alphabet of a free group

What do we mean by saying "a semigroup P is presented by generators and relations". Isn't it right only for the free semigroups? If it's right, we can't distinguish some two semigroups if they are ...
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1answer
57 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 ...
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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 & ...
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2answers
75 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 ...
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875 views

The compactness of the unit sphere in finite dimensional normed vector space

We define $ (\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $ \|.\|$ is defined to be any norm in $ \mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ ...
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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 ...
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1answer
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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: ...
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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 ...
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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 ...
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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 ...
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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 $d = \gcd(a,b)$.
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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 ...
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Reference request for tricky problem in elementary group theory

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of ...
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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.
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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 ...
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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 ...
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1answer
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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 ...
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Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
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1answer
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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 ...
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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!
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50 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?
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1answer
62 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 ...
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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 ...
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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 ...
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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 ...
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27 views

Foundation Semigroups Examples

I recently started looking into the subject of foundation semi-groups and I'm trying to find simple (none-trivial) examples of foundations semi-groups. I know that every locally compact group is ...
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Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
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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 ...
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Cayley's Theorem for Semigroups

I've read and fully understand Cayley's theorem for groups, however when I get to the theorem for semigroups I come to a complete stop. I've figured that the identity and cancellative properties are ...
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Is every sufficiently large semigroup $S$ a subsemigroup of the transformation semigroup on $S$?

In this question, all sets are finite. A semigroup is a set equipped with an associative binary operation, which I will call $\circ$. The tranformation semigroup on a set $X$ is the semigroup of maps ...
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1answer
149 views

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 ...
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If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?

This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...
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29 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 $C$. ...