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Questions tagged [semigroups]

A semigroup is an algebraic structure consisting of a set together with a single 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/universal algebra. Please use the more specific tag (semigroup-of-operators) whenever appropriate.

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, and semigroups with the notion of inverses are known as regular semigroups and inverse semigroups.

Semigroups are used in various areas of mathematics. $C_0$-semigroups are important in partial differential equations. Semigroups have also connections to automata theory.

Topological (and left/right topological) semigroups are also studied. Perhaps the best know result in this area is Ellis-Numakura lemma. Using Ellis-Numakura lemma, existence of idempotent ultrafilters can be shown.

1001 questions
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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 $X\to X$ with function composition as the…
Colin McQuillan
  • 6,717
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A cyclic subsemigroup of a semigroup S that is a group

I came across this problem while reading some lectures about semigroups here Lecture Notes on Semigroups by Tero Harju. Page $10$. He named it a nontrivial exercise. Let $r$ be the index and $p$ the period of an element $x\in S,$ a semigroup.…
Hassan Muhammad
  • 4,282
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Does there exist a semigroup such that $(xy)^n = x^n y^n$ that is non-Abelian? If so, can this property be finitely axiomatized?

Suppose $S$ is a semigroup such that for all $x,y \in S$ and all natural $n$ we have $$(xy)^n = x^n y^n.$$ If $S$ group, then it is Abelian; indeed a stronger statement holds, see here. Does there exist a semigroup with this property that fails to…
goblin GONE
  • 67,744
4
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$\mathcal{L}$ Green relation vs $\mathcal{H}$ Green relation

Let $S$ be a regular semigroup. The Green's equivalence relation $\mathcal{L}$ is defined on $S$ as follows. $$s\mathcal{L}t \mbox{ if and only if } Ss=St.$$ Similarly using cyclic right ideals of $S$, the Green's relation $\mathcal{R}$ is defined.…
khers
  • 379
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How to prove a semigroup with properties $x^3=x$ and $x^2y^2=y^2x^2$ is commutative

If $S$ is a semigroup such that $$(\forall x,y\in S)\quad x^3=x\quad\text{and}\quad x^2y^2=y^2x^2,$$prove that $$(\forall x,y\in S)\quad xy=yx.$$ All I did is prove $$x^2y^2=(x^2y^2)^2\quad\text{and}\quad xy^2x=(xy^2x)^2.$$ Proof of…
bEtAVs
  • 79
4
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Second isomorphism theorem for semigroups

Let $S$ be a semigroup. If $S$ has a zero element, define $S^0 = S$. Otherwise define $S^0$ as $S$ with a zero element adjoined. A subset $I$ of $S$ is an ideal if it is closed under left and right multiplication by every $s \in S$. The Rees…
Ris
  • 1,292
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Is every semigroup with (possibly non-unique) division a group?

Let's say that a semigroup $(S,\cdot)$ has weak division if for all $a,b\in S$ there exist $c,d\in S$ such that $ac = b$ and $da = b$. Note that we don't require $c$ and $d$ to be unique. This property may already have a name in the semigroup…
Alex Kruckman
  • 76,357
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Evans' theorem of embeddings into 2-generator semigroup.

I recently found this theorem when I am studying Semigroup presentations. Theorem: If S is a semigroup generated by denumerably many elements, then S can be embedded into a semigroup generated by two elements. As a hint for constructing the proof…
Hassan Muhammad
  • 4,282
3
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Proof that natural numbers with multiplication are cancellative

i want to show that the multiplicative semigroup of the natural numbers without zero are cancellative, that is: For all $h,x,y$, $hx=hy \implies x=y$ I tried proving this by induction over $h$ and letting $x,y$ fixed. For $h=1$ it is trivial. But…
Blue2001
  • 349
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Is there a semigroup which admits no involution?

A semigroup with involution is a semigroup $(S;*)$ equipped with a unary function $f$ such that $f(f(x))=x$ and $f(x*y)=f(y)*f(x)$. I want to know, does there exist a semigroup for which there exists no corresponding involution? I would be…
user107952
  • 20,508
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Show $\mathcal{R}$ related

If $\alpha\in T_n$ and $\beta\in S_n$ show that $\alpha \mathcal{R} \alpha\beta$ ($T_n$ is the full transformation monoid, and $S_n$ is the symmetric group, both on $\{1,2,\ldots ,n\}$). Does this mean show $\alpha T_n = (\alpha\beta )S_n$? How…
Sam
  • 41
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Semigroup over a finite set that is nothing more than that

Ok so a semigroup only has associativity. Suppose i have a set $S = \{a,b,c,d\}$ Someone give me an example for the function $+:S \times S \rightarrow S$ (in table form) so that is only associative so that $(S, +)$ is just a semigroup and nothing…
userSd93r0294029480294850934t0
  • 135
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Semigroup congruence defined on generators

Let $S$ be an abelian semigroup generated by a nonempty set $X$. Suppose $\sim$ is an equivalence relation on $S$ satisfying the condition: "If $x \sim x'$ and $y \sim y'$ then $x+y \sim x'+y'$ for all $x, x', y, y' \in X$." Does it follow that…
JMD
  • 51
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1 answer

Coinciding the ideals in a semigroup

Kindly, I am asking to light my mind by some leading hints: Can minimal and maximal ideals in a finite non-commutative semigroup $S$ coincide? Thanks for the time!
Mikasa
  • 67,374
2
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Prove that a cyclic semigroup is either finite or isomorphic to $\langle \mathbb{N},+ \rangle$

A semigroup $G$ is cyclic if $G$ is generated by a single element. I know that a finite cyclic group generated by $a$ is necessarily abelian, and can be written as $\{1, a, a^2, . . . , a^n−1\}$ where $a^n = 1$, or in additive notation, $\{0, a,…
fahimeh choobtarash
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