# 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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### One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
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### An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
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### Period of semigroup

Let $S$ be a finite semigroup of order $n$. Suppose that $S$ has index $m$ and period $r$, i.e. $S$ satisfies the identity $x^{m+r} = x^m$. Then it is quite easy to show that $m \leq n$. My question ...
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### About automorphisms of commutative semigroups

Suppose that $M$ is a commutative monoid and that the product $P$ of $M$ and the nonnegative integers $\mathbb{N}$ with addition has no nontrivial automorphisms. The set $S$ of pairs $(m,n)$ in $P$ ...
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### Finite Linearly Ordered Abelian Monoids

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids". The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The ...
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### Terminology for idempotents that commute with every other idempotent

Given a semigroup $S$, is there terminology for those $x \in S$ such that the following hold? $x$ is idempotent Given any idempotent $y \in S$, we have $xy=yx$. Comments. Let $E$ denote the set ...
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### Help in this characterization of the gaps of the symmetric numerical semigroups

Before my question, some background: Definition 1: A numerical semigroup is a subsemigroup $N$ of the additive semigroup $\mathbb N$ of the non-negative integers such that $\mathbb N-N$ is ...
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### Who first studied semilattices?

Historically, who first studied semilattices, as opposed to lattices or Boolean algebras? (With or without identity, I do not mind.)
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### Example of a semigroup with unique idempotent which is not a monoid

I am searching for an example of a semigroup, with unique idempotent element, that is not a monoid. Please help.
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### The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
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### Reference request on numerical semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
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### A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
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### If $G,H$ are monoids and not groups, prove that $f(e_G)=e_H$ may be wrong.

Problem: If $f:G\longrightarrow H$ is a homomorphism of groups, then $f(e_G)=e_H$ and $f(a^{-1})=f(a)^{-1}, \forall a\in G$. Show by example that the first conclusion may be false if $G, H$ are ...
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### About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
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### How many associative ternary operations there are on a finite set?

We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$ If $X$ is a set and and cardinality is $|X|=n$ then ...
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### Example of an inverse semigroup

An inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...
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### Evolution Equation

Let $X=L^2(0,\pi)$. Define the operator $(A,D(A)$ by: $$D(A)=\{u\in H^2(0,\pi):u(0)=u'(\pi)=0\} ,\quad \quad Au=u''$$ Show that $A$ is the infinitesimal generator of a $C_0$ semigroup of contractions ...
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### Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
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### Commuting probability

http://www-rohan.sdsu.edu/~vadim/commute.pdf Where does the commuting probability formula on page 8 come from ?
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### operator semigroups with negative growth bound.

Let $(T(t))_{t\geq 0}$ be a strongly continuous semigroup on a Banach space $X$, for which we assume that its growth bound $\omega_0$ is negative. Let $(A,D(A))$ be the generator of $(T(t))_{t\geq 0}.$...
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### Understanding the definition of a Integral

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on $X$ ...
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### Is Writing a Semi Group?

While writing this question, I compose letters from a set $L=\{a,...z,A,...Z\}\cup\{\;\text{ } \;\}$. Writing has a binary operation which is associative. The result always is an element of $L^n$. ...
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### Proof of no identity element in a multiplication semi group

I have a question from my workbook: Let $\mathrm{E}$ denote the set of even integers. This forms a semi-group under multiplication. Show that there is no identity in this semi-group. Now this is ...
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### An intersection equation in semigroups

Do you have an example of a semigroup $S$ and a collection of its subsets $(A_i)_{i\in I}$ and $a\in S$ such that $$a\big(\bigcap_{i\in I}A_i\big)\ne\bigcap_{i\in I}aA_i$$ ?
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### Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
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### Involution on inverse semigroups

I'm trying to prove the following for inverse semigroups $\bf Def:$ an inverse semigroup $S$is a semigruop such that for each $x\in S$ the exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. An ...