# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### The code SetReducedMultiplication for semigroups

Let $S$ be a finite semigroup like this: ...
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### 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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### 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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### 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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### 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.
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 ...