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.

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Semigroups and relations

Recall that if $S$ is a semigroup then for $a\in S$ $$Sa = \{sa : s \in S \}\text{,}\;\;\;S^1 a = Sa \cup \{a\}\text{,}$$ $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The ...
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Rings and idempotent semirings

If $\mathbb{R}$ is the real numbers and x#y=max{x,y} then ($\mathbb{R}$,#,+) is a semiring where ($\mathbb{R}$,#) is a semigroup and + distributes over #. If you have a set R with three distinct ...
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Quasicommutative semigroups.

A Semigroup is called quasicommutative if for all elements $a,b$ there is some $r≥1$ such that $$ab=b^ra$$ We know that every commutative semigroup is also quasicommutative, so we can make lots ...
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Properties of Group $\to$ Monoid $\to$ Semigroup.

In How to get a group from a semigroup Arturo Madigin writes: So if we look at the composition of the forgetful functors $\bf Group \to \bf Monoid \to \bf Semigroup$, we obtain a right adjoint by ...
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Weak generator of Feller semigroup

Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in ...
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A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
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Schrödinger Kernels on manifolds

Let $M$ be a compact Riemannian manifold and $\Delta$ be the Laplace-Beltrami operator. It is well-known that the solution operator to the heat equation $e^{t \Delta}$ is smoothing for $t>0$ and ...
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Definition of tautological action

What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions? For reference the particular sentence is: "A ...
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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 ...
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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 ...
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Semigroup question

I am looking for the technical term for an element of a transformation semigroup that sends everything to one state. The best term I could think up was filter. For those that don't know a ...
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hint request: counting the elements in a subset of a numerical semigroup

I'm working on a problem for an online judge and I'm stuck. I'd like a nudge in the right direction (not an outright solution, please), relevant references, theorems, etc. After reading through ...
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Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...
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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 ...
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Is this a series convergence criterion? (Related to the Hilbert-Waring theorem)

By the set of natural numbers I will mean $\mathbb N=\{n\in\mathbb Z\,|\,n\geq0\}.$ I have come across a condition on a sequence $(a_n)_{n=0}^\infty$ of natural numbers that I feel may imply that ...
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Find identity element, invertible and inverses in $T=\mathbb Z \times \mathbb Q$

Let the following operation be defined on $T=\mathbb Z \times \mathbb Q$: $$\begin{aligned}(a,b)\centerdot (c,d) = (-ac, b+d+2) \end{aligned}$$ in the commutative semigroup $(T, \centerdot)$, find ...
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Question on Malcev's _Immersion of an Algebraic ring into a skew field_.

I'm reading the paper Immersion of an algebraic ring into a skew field by Malcev. Doi: 10.1007/BF01571659, GDZ. On the third page of the paper, he writes that If $\alpha\beta\sim\gamma\delta$ and ...
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Tell if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it

Let the operation $\odot$ be defined in $\mathbb Z_6$ as follows: $$a \odot b = a +4b+2$$ check if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it. This is the way I ...
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What should I call this commutative monoid of order three?

I'm looking for a name for the monoid given by the following table: $$ \begin{array}{c|ccc}&1&a&b\\ \hline 1&1&a&b\\ a&a&1&b\\ b&b&b&b \end{array} $$ ...
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Is there a name for a semigroup whose idempotents form a subsemigroup?

For a semigroup $S,$ I will denote by $E(S)$ the set of all idempotents of $S$. For $X\subseteq S,$ let $X^2$ mean $\{xy\,|\,x,y\in X\}.$ Is there a name for the class of semigroups $S$ such that ...
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What are the subsemigroups of $(\mathbb N,+)?$

While trying to solve a somewhat bigger problem, I realized that I don't know what the subsemigroups of one of the most important semigroups, $(\mathbb N,+)$, are. (I assume $0\not\in\mathbb N$.) I've ...
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Is there a name for a subset $S$ of a group or a semigroup such that every two elements of $S$ commute?

Let $G$ be a group and $S$ its subset. I would like to consider the following condition on $S$. For every $x,y\in S,$ we have $xy=yx.$ This is trivially equivalent to $S\subseteq C(S).$ The ...
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Is the universal inverse semigroup of a commutative semigroup an embedding?

The question of existence of a universal inverse semigroup of an arbitrary semigroup has been answered before (this is a construction similar to the Grothendieck group). Let's refer to the universal ...
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What are some (or even one) interesting examples of (non-group) semigroups?

I'm going to give a lecture on Alon and Schieber's Tech Report on computing semigroup products (Optimal Preprocessing for Answering On-Line Product Queries). Basically, given a list of elements ...
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If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?

This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...
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What condition on $m$ turns this semigroup into a monoid?

Suppose you have a monoid $(M,p,1)$ (viewing it as a triple of a set $M$, operation $p$, and unit $1$). Then for some $m\in M$ we can define a new product $p_m$ in $M$ by $p_m(a,b)=amb$. It's easy to ...
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Semigroups of matrices with zeroes and a single 1

I stumbled upon this while reviewing a Harvard lecture on abstract algebra. What I want to know is if these semigroups are known and, if so, what they are called. I've checked the assertions below for ...
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If every element of a monoid is right invertible, then every element is invertible

Let $G$ be a set with associative binary operation and a unit. Assume that for every $g\in G$ there exists $x \in G$ with $gx = 1$. Prove that $xg = 1$ is a consequence.
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Is the additive semigroup of natural numbers the multiplicative semigroup of a ring?

$\mathbb N$ will denote the set $\{0,1,2,\ldots\}.$ The semigroup $(\mathbb N,+)$ doesn't have a zero element. $\mathbb N^0$ will denote the semigroup $\mathbb N$ with zero adjoined, that is the set ...
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If cancellation laws hold, then a finite semi-group is a group [duplicate]

Show that if both cancellation laws i.e $w.a = w.b \implies a = b$ and $a.w = b.w \implies a = b$ holds then a finite semi-group (a finite set with associative binary operation) is a group. I have ...
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Why isn't there much study on right groups (left groups)?

This question is related to and inspired by the question Why are groups more important than semigroups?. I am curious why I don't see much studies on right groups. On Pp.37 of Clifford& ...
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Errata for Semigroups and Combinatorial Applications by G. Lallement?

I am asking this question after years of frustration with the typos in the subject book I have read. It has been cited and referenced by many literatures and books in math and computer science. ...
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Constructing all semigroups over a given set without backtracking

Is there a procedure to construct all semigroups over a given set without backtracking? (Edit: see also how many associative binary operations are there on a finite set and ratio of semigroups over a ...
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Group identities and inverses

A set is a set. A magma is a set with a binary operator. A semigroup is a magma with an associative binary operator. A monoid has a two-sided identity. And a group has two-sided inverses. I am ...
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give a counterexample of monoid

$G$ is a monoid,$e$ is its identity,if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$. if not,please prove $b=c$ thanks a lot.
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Two elementary question on automaton and language

1.What is the definition for a semigroup(or monoid) recognizing a set of words(or language)?2.Are recognizable,rational and regular equivalent to each other with respect to a language? PS:The reason ...
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How to ensure the syntactic semigroup of $X$ is the smallest semigroup recognizing $X$

Show that the syntactic semigroup of $X$ is the smallest semigroup recognizing $X$ in the sense that, for every semigroup $S$ recognizing $X$, there exists a morphism from $S$ onto the syntactic ...
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Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?

This question is a by-product of this one. I'm asking it because of this comment by Tara B. I'll repeat the definitions. The full transformation semigroup $\mathscr T_X$ on a set $X$ is the semigroup ...
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Idempotent separating congruence of an inverse semigroup.

Please can sombody help me with the proof of this lemma, or even a construction of the proof? I will be glad for that. Lemma: Show that if $\rho$ is an idempotent separating congruence of an ...
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Description of Green's relations $\mathcal{L}$, $\mathcal{R}$ in an inverse semigroup

Theorem: Let $S$ be an inverse semigroup, and let $x,y\in S$ and $e,f\in E_{S}$ then $x\mathcal{L}y$ if and only if $x^{-1}x=y^{-1}y$ $x\mathcal{R}y$ if and only if $xx^{-1}=yy^{-1},$ where $E_S$ ...
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Is there a topology on the full transformation semigroup?

$\mathscr T_X$ will denote the set of all functions from a non-empty set $X$ into itself, with the binary operation of composition $\circ$ making it a semigroup, called the full transformation ...
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$\mathcal{R}$-relation in full transformation semigroup

Let $T_{X}$ be the full transformation semigroup on $X$. For $\alpha$, $\beta \in T_{X}$ $$\alpha \mathcal{R}\beta \text { if and only if there exist }\gamma,\gamma' \in ...
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Do endormorphisms of a semigroup form a monoid? Automorphisms form a group?

I came across this theorem when studying thess lecture notes Theorem: Prove that The endomorphisms of a semigroup S form a monoid. The automorphisms of a semigroup S form a group. I do not ...
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What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
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Ring without distributive law?

I recently came across a binary operation (in a very non-algebraic context - it's a way to organize a certain updating of log-likelihood-ratios) and was idly wondering whether it is any kind of ...
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Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
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Must a pseudoinverse of a von Neumann regular element be regular?

Background Let $R$ be a ring or a semigroup. We say that $x\in R$ is a von Neumann regular element of $R$ if there exists $y\in R$ such that $$xyx=x.$$ Any $y\in R$ satisfying the above equation is ...
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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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If the product of two idempotents is idempotent, must the two idempotents commute?

It is a basic fact that when two idempotents $e,f$ in a semigroup $S$ commute, then $ef$ is an idempotent. Is the converse true? Is it true for idempotents in rings?
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Evan's 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 denummerably many elements, then S can be embedded into a semigroup generated by ...