# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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