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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.

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Semigroups (ideals of a semigroup)

How many ideals are there in the $\mathbb Z_{28}$? $\mathbb Z_{28}=\{0, 1, 2, 3, 4, 5, ..., 27\}$ is a semigroup under multiplication modulo 28.
Tom young
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Semigroup Presentations

I am reading a text on semigroups and encountered the following: Consider the semigroup $S$ defined by the presentation $\langle a_1,a_2,...,a_n|a_i^2=a_i, a_ia_j = a_ja_i(1\leq i,j\leq n)\rangle$. By definition, every element $s \in S$ is a product…
Luke
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A question about right ideals in semigroups and the complementary operation

Let $S$ be a semigroup and $X$ be a non-empty subset of it. Note that for a right zero semigroup $S$ and every $x\in S$ we have $xS=S$. So for a proper non-empty subset $X\subset S$ we have $S=\cup_{x\in X}xS=\cup_{y\in S\setminus X}yS$. On the…
khers
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A basic question about finite semigroup

Let $S$ be a finite semigroup. Recall that every element $a\in S$ determines a unique pair of positive integers $\iota=\mathrm{ind}(a)$ and $\rho=\mathrm{per}(a)$, called the index of $a$ and the period of $a$, respectively. These are the smallest…
boaz
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Period of an element in direct product of finite semigroups

Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the period of $(x,y)$ ? I know that if $$\mathrm{index}(x)=\mathrm{index}(y)$$, then the period of $(x,y)$ is $$\mathrm{lcm(period}(x),\mathrm{period}(y))$$. What can we say…
boaz
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An exponent of a finite semigroup

Let $S$ be a finite semigroup. I want to define an exponent of $S$ which is a generalization of the familiar term in finite groups. Recall that every element $a\in S$ determines a unique pair of positive integers $\iota=\mathrm{ind}(a)$ and…
boaz
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Free band on $n$ generators is a semilattice of rectangular bands.

Exercise (part of it) comes from Clifford, Preston, Algebraic theory of Semigroups. By $\mathbb{FB}_n$ we denote the free band on $n$ generators, that is, a free semigroup generated by a set $X=\{x_1,...,x_n\}$ subject to relations $w^2 = w$ where…
Jakobian
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A regular quasicommutative semigroup $S$

Here is a theorem: I could go inside the theorem and know some few points of it. It's told that If all elements of $H_e$ are of finite order so the group $H_e$ is Hamiltonian. My question is how is it possible? In fact, if $H_e$ has an element…
Mikasa
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Example of a quotient semigroup which can not be embedded into the finite semigroup

For $S=(S,\cdot)$ a finite semigroup and $\sim$ a congruence relation on $S$, we have a quotient semigroup $(S/{\sim}, \cdot)$ with the operation as $[s_1]\cdot[s_2]=[s_1\cdot s_2]$. $S \rightarrow S/{\sim}$ defined by $s \mapsto [s]$ forms a…
Anuj More
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The Baer-Levi semigroup is right simple and right cancellative, but neither left simple nor left cancellative.

This is Exercise 2.6.5(b) of Howie's "Fundamentals of Semigroup Theory". The Details: (Here I write maps on the right of their arguments, so $X\alpha$ means $\alpha(X)$.) Let $S$ be a semigroup. Definition 1: Let $X$ be a countably infinite set and…
Shaun
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Show that $\mathcal L ⊆\mathcal L^*$ on a semigroup $S$

Define $\mathcal L^*$ by $a \mathcal L^* b$ if and only if $∀x,y \in S^1$, $ax=ay \iff bx=by$ Show that $\mathcal L ⊆\mathcal L^*$ on a semigroup $S$ This seems like it should be a very simple question but it isn't working out as I would have…
user2973447
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Semigroup with exactly one left(right) identity?

Are there any examples of a semigroup (which is not a group) with exactly one left(right) identity (which is not the two-sided identity)? Are there any “real-world” examples of these (semigroups of some more or less well-known mathematical objects)…
Artem Pelenitsyn
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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 bijection goes from $H_b$, but don't see how it…
Mark
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L,R,H,D,J relations on a completely simple semi group represented my a rees matrix

I am trying to tackle the following semigroup question. I can't see why my answer is wrong but I haven't used the fact the semigroup is COMPLETELY simple anywhere so I think there must be an error somewhere in my proof. Any advice? Question Consider…
ENAFMTH
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If $S$ is a finitely generated periodic semigroup with the permutation property, then $S$ is finite.

In A. Nagy's Special Classes of Semirings, the first theorem is: Theorem 1.1 A finitely generated semigroup is finite iff it is periodic and has the permutation property. The definitions are as follows. Let $S$ be a semigroup. Call $x \in S$ is…
goblin GONE
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