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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Generator of an analytic semigroup with a compact resolvent --> pairwise conjugate eigenvalues?

I am reviewing a paper in which the authors claim that their operator has eigenvalues $\lambda$ that are either real or pairwise conjugate (meaning that if $\lambda$ is an eigenvalue, then also the ...
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160 views

Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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The Fundamental Semigroups of a Topological Object?

I had the following idea for a generalization of the "fundamental group" of a manifold. So the idea basically was that we can consider a manifold $O$ which has boundary $\partial O$, and instead of ...
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Irreducible representations of locally compact semigroups

What class of semigroups have finite dimensional irreducible representations. For example, If we have a compact groups $G$ then every continuous irreducible representation of $G$ is finite ...
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11 views

Finding representation in numerical semigroup

I'm given $(n_1,n_2,n_3)$, with $\operatorname{gcd}(n_1,n_2,n_3)=1$. Then, I need to find $c_1$, the least positive integer such that $c_1n_1=n_2\mathbb{N}+n_3\mathbb{N}$. I additionally need the ...
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Does associativity imply closure?

Does associativity of binary operation imply closure under this operation? Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes ...
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97 views

Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

[Update: I've now asked the same question on mathoverflow.] For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now suppose ...
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Limit relevant to parametrised semi-group 2

Let $s\geq 1, \epsilon >0, T>0$ and $f \in \mathcal{C}([0,T], H^s(\mathbb{T}))$. Define the function $$g(t,x):= \int_0^t(Id-\exp\left(-i\tau\epsilon \Delta)\right)f(\tau,x)d\tau.$$ I want to ...
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70 views

What is the smallest variety containing all monoids and all semigroups with a one-sided zero?

What is the smallest variety (in the universal algebra sense) containing all monoids, all semigroups with a left zero, all semigroups with a right zero, and as few other models as possible? So far, ...
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15 views

The Krull-Schmidt-Remak Theorem for Semigroups and Monoids

For finite groups, the Krull-Schmidt-Remak-Theorem holds, i.e. if $$ H_1 \times H_2 \times \ldots \times H_k \cong G_1 \times G_2 \times \ldots \times G_l $$ where the $H_i, G_i$ could not be further ...
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42 views

Ultrafilter invariant semigroups

This may be quite basic but I was unable to figure out the answer on an uncountable set. Suppose that $S$ is an infinite set and $U$ a non-principal ultrafilter on $S$. Does there exist a commutative ...
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69 views

Where is the notion of anti-isomorphism useful

Let $(S,\cdot)$ and $(T,\circ)$ be semigroups (or some algebraic structure with an operation), then they are anti-isomorphic if there exists some $\varphi : S \to T$ such that $$ \varphi(xy) = \...
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30 views

Semigroups and Finite Dimensional Representaions

It is well known that if a group $G$ is compact then every irreducible continuous representation of $G$ is finite dimensional. As far as I know the semigroup equivalent of this statement is not true. ...
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1answer
35 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 \...
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1answer
25 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, $\...
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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"), (...
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1answer
24 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 ...
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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
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 ...
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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 ...
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37 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 ($\...
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33 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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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?
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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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91 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 ...
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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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1answer
77 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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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 \...
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1answer
91 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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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 ...
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1answer
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The code SetReducedMultiplication for semigroups

Let $S$ be a finite semigroup like this: ...
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88 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 x^2y^2=y^2x^2=(...
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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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80 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: ...
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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
68 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 $x$...
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1answer
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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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1answer
35 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 ...
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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 t\...
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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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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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36 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 ...
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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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1answer
35 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
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 \...
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1answer
65 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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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 1....