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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Equational theories in commutative idempotent monoids

It might well be that my question is trivial but I'm not a mathematician, I just need a formalization for algebraic semiotics. If I have a commutative idempotent monoid, I can define a partial ...
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2answers
105 views

Building factor semigroup w.r.t. $I_n = \{ w \in A^+ : |w| \ge n \}$.

I don't understand the following factor semigroup. Consider the pseudovariety $N$ of nilpotent semigroups. For any finite alphabet $A$, let $I_n = \{ w \in A^+ : |w| \ge n \}$. Then $A^+ / I_n$ is ...
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1answer
79 views

Proving $\mathcal{D}$-class of semigroup.

Theorem 2.17 If $a$ and $b$ are elements of a semigroup $S$, then $ab\in R_a\cap L_b$ if and only if $R_b\cap L_a$ contains an idempotent. If this is the case, then $$aH_b = H_a b = H_a H_b ...
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Semigroup which satisfied $(xy)^{\pi} = (xy)^{\pi}x$ and principal right ideals

Let $(S,\cdot)$ be a finite semigroup, then every $s \in S$ has an unique idempotent power, i.e. there exists a smallest $i \in \mathbb N$ such that $s^i$ is idempotent. It is the unique idempotent in ...
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1answer
227 views

Recommend a semigroup theory book.

I have learnt semigroup theory and I want some good book for studying by myself. Thank you everyone.
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1answer
66 views

Please check my solution, inverse semigroups.

Let $x$ and $y$ be elements of an inverse semigroup $S$. Then the following are equivalent: (1) $xy^{-1}x = x$ (2) $x^{-1}yx^{-1}=x^{-1}$ I'm not sure in my solution, ...
7
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3answers
649 views

Is a semigroup $G$ with left identity and right inverses a group?

Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group? Now, If both the identity and the ...
8
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1answer
130 views

Structure theorem for finitely generated commutative $semi$groups.

$\newcommand{\ZZ}{\mathbb{Z}}$ It is a classical theorem that a finitely generated commutative group is isomorphic to one of the form: $$ \ZZ^n \oplus \ZZ/{m_1}\ZZ \oplus \ZZ/{m_2}\ZZ \oplus \dots ...
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2answers
159 views

Whether $L=\{(a^m,a^n)\}^*$ is regular or not?

I am condidering the automatic structure for Baumslag-Solitar semigroups. And I have a question. For any $m,n \in Z$, whether the set $L=\{(a^m,a^n)\}^*$ is regular or not. Here a set is regular means ...
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1answer
205 views

Help me prove equivalently of regular semigroup and group.

Let $S$ be a semigroup. Prove that the following are equivalent: $\forall a \in S \exists! x \in S$ such that $ax \in E(S)$ where $E(S)$ is the set of all idempotent. $\forall a \in S \exists! x \in ...
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74 views

Is there a standard name for this semigroup?

Given a semigroup $X,$ we can form a new semigroup $Y$ by asserting that: the carrier of $Y$ is the set $X^2$, and the law of composition in $Y$ is given by $(a,b)(a',b')=(aa',b'b).$ Finally, ...
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1answer
58 views

semigroups-terminology question

Let the element $x$ of a commutative semigroup has the property $x=a+b\Rightarrow a=x~$or $b=x$. I call such an element "prime". My question is: what is the right term about such elements. ...
4
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1answer
240 views

Structure Descriptions (GAP) in semigroups

As can be easily seen through GAP manual: 37.6 Structure Descriptions StructureDescription( G ) A The method for StructureDescription exhibits the ...
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1answer
151 views

Is closure of a semigroup again a semigroup?

Let $S$ be a compact left-topological semi-group (meaning, $S$ is both a semi-group and a compact Hausdorff topological space, and the map $x \mapsto x y$ is continuous for any fixed $y$, but the map ...
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2answers
35 views

Size of the image (rank) in $T_n$

For $\alpha, \beta \in T_n$ (full transformation semigroup/monoid - set of all maps from $\{1,2,\ldots, n\}$ to itself), show that $|\text{Im}(\alpha\beta)| \leqslant |\text{Im}(\alpha)|$ and ...
3
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2answers
57 views

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 ...
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0answers
65 views

irreducible words in a semigroup

Let $S$ be a monoid generated by $\{x_1,x_2,x_3,x_4,x_5\}$ with the following relations: $x_i^2=0$ for all $i$, $x_ix_j=x_jx_i$ for all $|i-j| \geq 2$ and $x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}$ for all ...
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0answers
57 views

Trivial question about semigroups

I have a question about analytic semigroups. I know that if $A: D(A)\subset X\to X$ is a sectorial operator on $X$, it generates an analytic semigroup $e^{tA}$. One of the properties of this semigroup ...
3
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1answer
109 views

Does someone know other examples of $K$-semigroups?

In our work on projective representations we need use the following object: Let $K$ be a field. By a $K$-semigroup we mean a semigroup $S$ with $0$ and a map $K \times S \to S$ such that $\alpha ...
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61 views

Spaces of class $J_\alpha$

This question is about the spaces of class $J_\alpha$. Given three Banach spaces $Z\subset Y\subset X$ (with continuous embeddings), and given $\alpha\in (0,1)$, we say that $Y$ is of class $J_\alpha$ ...
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0answers
32 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
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1answer
736 views

Semigroups, monoids, & groups!

I need help determining if these are semigroups, monoids, or groups? a) $\mathbb Z ^+$, where $\#$ is defined as ordinary multiplication b) $\mathbb Z ^+$, where $a \# b$ is defined as $\gcd(a,b)$ ...
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1answer
85 views

semigroup presentation and Diamond lemma

Suppose a semigroup (possibly infinite) presentation is given with generating set $S$ and relations $R$. I need to prove using Bergman's diamond lemma that the semigroup is non-zero i.e, I have to ...
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3answers
111 views

Semigroup generators and Cones

In $\mathbb{R}^2$, let $C$ be the cone (non-negative linear combinations) on the two vectors $$ v_1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad v_2=\begin{pmatrix} 2 \\ 3 \end{pmatrix} $$ Consider ...
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5answers
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Is there an idempotent element in a finite semigroup?

Let $(G,.)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
2
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1answer
103 views

Finitely generated semigroup gets the finitely generated subsemigroup?

From N.RUSKUC's paper "On Large Subsemigroups and Finiteness conditions of Semigroups", there is a theorem, Here large subsemigroup means $S$\ $T$ is finite. In this side "=>" of the proof in the ...
4
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2answers
72 views

How to move from a right semigroup action to a left semigroup action?

Let $S$ be a semigroup and $X$ any set. Define a left action of $S$ on $X$ to be a map $\sigma: S \times X \rightarrow X$ with the property that $(st)x = s(tx)$, where we define $gx = \sigma(g,x)$ ...
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How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
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117 views

How are the powers being changed

I have a semigroup $S$ including a generator, say $d$, such that $$d^4=d$$ I am trying to guess the general rule of $d$'s powers such that when I want to calculate $d^n, n\in\mathbb N$; I can simplify ...
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4answers
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Inverse elements in the absence of identities/associativity.

Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows. $(xy)z=x(yz).$ $xe=ex=x.$ ...
3
votes
1answer
137 views

Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
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3answers
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Proving a binary operation on $\mathbb{Z}$ gives a semigroup

Let $x\circ y= x +y-xy, \quad (x,y) \in \mathbb{Z}$ where $\circ$ is a binary operation on $\mathbb{Z}$, prove that this is a semigroup. My Work To prove we have to check two things: $\mathbb{Z}$ ...
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2answers
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In a semigroup, $ab=ba\Rightarrow (ab)^k=a^kb^k$.

Let $a,b \in S$ where $(S, *)$ is a semigroup and suppose $ab = ba$. Prove that $(ab)^k = a^kb^k$ for all $k ≥ 1$
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1answer
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inverse on topological semigroup

Assume $(G,\cdot)$ denotes a topological semigroup (no id, non-commutative). Let $V\subset G$ be open and take some arbitrary $g\in G$. Define $$gV^{-1}:= \bigcup_{x\in V}gx^{-1} = \bigcup_{x\in ...
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Unitary operators - convergence problem

Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$. Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
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2answers
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$\mathcal{D}$-classes

Let $$\alpha = \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}} 1&2&3\\ 1&3&3 \end{array}\right) \in \mathcal{T}_3\text{.}$$ (a) Show that the ...
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0answers
135 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
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Name for a function whose image has smaller cardinality than its domain

I asked this question in the comments of this question, whose title would have done just as well for mine. But I suppose it should be a separate question. Is there a name for functions ...
3
votes
1answer
215 views

Rees matrix semigroup

Let $\mathcal{M}^0 = \mathcal{M}^0 ( G; I , \Lambda ;P)$ be a Rees matrix semigroup ($G$ a group, $I$, $\Lambda$ non-empty sets, $P=(p_{\lambda i})$ a $\Lambda \times I$ matrix over $G \cup \{0\}$ ...
2
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1answer
67 views

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 ...
2
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1answer
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Semigroup homomorphism and the relation $\mathcal{R}$

Let $S$ be a semigroup and for $a\in S$ let $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The relation $\mathcal{R}$ on a semigroup $S$ is defined by the rule: $$a\;\mathcal{R}\; ...
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1answer
155 views

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 ...
3
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1answer
140 views

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 ...
8
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1answer
229 views

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 ...
9
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1answer
274 views

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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1answer
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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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0answers
61 views

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 ...
4
votes
1answer
219 views

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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1answer
62 views

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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1answer
83 views

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 ...