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.
957
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Show that a semi-group (E,T) (I.e., with T associative) satisfying a certain property is a monoid (i.e., possesses a neutral element)
Let $E$ be a set with an internal operation $T$ associative such that there exist $a \in E$ such that :
$(∀y\in E) (\exists x\in E) \ y=aTxTa$
Prove that $(E;T)$ has an identity element.
What I have ...
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0
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0
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22
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Markov semigroup: the mathematical definition of $\mathcal{L}(X_t | \mathcal{L}(X_0)=\nu)$
I'm reading about Markov semigroups from these slides, i.e.,
We consider a Markov process $(X_t)_{t \geq 0}$, with state space $E$, assumed to be ergodic with unique invariant probability measure $\...
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0
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0
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19
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Can a digraph in GAP be obtained as a virtual output?
In GAP when we draw a digraph corresponding to transformation semigroup using "Digraph" command we get "DigraphFromDigraph6string(" & CQ' g")". Is there a way to ...
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1
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1
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35
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Is multiplication by a fixed element an open map in a topological semigroup?
In a topological semigroup $G$, multiplying by a fixed element is continuous as we can decompose it into $$G \overset{\text{diagonal}}{\longrightarrow} G\times G \overset{\text{const}\times\text{id}}{\...
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0
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8
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When $\overline{Kx}= \overline{K}x$ for a semiflow $(T, X)$, where $K\subseteq T$, $x\in X$.
Let $T$ be a topological semigroup, $X$ be a topological space. Also let $\varphi:T\times X\to X$ is a continuous action.
I think if $K\subseteq T$ is a compact set, then $\overline{Kx}= Kx$, by ...
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4
votes
1
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81
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$\mathcal{L}$ Green relation vs $\mathcal{H}$ Green relation
Let $S$ be a regular semigroup. The Green's equivalence relation $\mathcal{L}$ is defined on $S$ as follows.
$$s\mathcal{L}t \mbox{ if and only if } Ss=St.$$
Similarly using cyclic right ideals of $S$,...
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2
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1
answer
29
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Non-trivial (regular) open semigroups of the open unit interval $(0,1)$?
This is a follow-up question to the one I asked here.
Namely, are there examples of open (or even regular open) subsets of $(0,1)$ which are multiplicative subsemigroups of $(0,1)$ but are not of the ...
- 455
2
votes
3
answers
75
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Non-trivial semisubgroups of the unit interval? [closed]
The open unit interval $(0,1)\subset \mathbb{R}$ forms a semigroup under multiplication. What are examples of subsemigroups of this semigroup, which are not intervals of the form $(0,a)$ for $a\in (0,...
- 455
1
vote
1
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46
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Find the energy of a partial differential equation
Consider the following equation
$$
y_{tt} - y_{xx} - y_{tx} - y_{txx} = 0 , \ \ (x,t) \in (0,1) \times (0,\infty)
$$
with $y(0,t) = y(1,t) = 0$, $t \in (0,\infty)$. I want to find the energy of system....
0
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0
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16
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Reference for a result in regards to semigroup theory
I'm looking for a reference for the following result:
Let $(S(t))_{t\geq0}$ be the analytic semigroup of contractions, generated by the Laplace operator $\Delta,$ with homogeneous Neumann boundary ...
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0
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0
answers
10
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Is there any result that makes me conclude that $(i\lambda I - A)^{-1}$ is continuous?
Let $X = H_{0}^{1}(-1,1) \times H_{0}^{1}(-1,1) \times L^{2}(-1,1) \times L^{2}(-1,1)$. Consider $\lambda > 0$ and a$A$ a dissipative operator generates a $C_0$−semigroup of contractions. Assume ...
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0
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16
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Simplifying intergral over $C_0$ semigroup
Let's say $(P_t)_t$ is a semigroup on $C_0(\Bbb R)$ with a generator $A$ and consider $J = \int_0^T P_t\mathrm dt$. Simply symbolically we get that
$$
J = \int_0^T \mathrm e^{At}\mathrm dt = A^{-1}(\...
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7
votes
1
answer
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Prove $x^3 = x^2$ in an abelian semigroup given two elements with a specific property
Let $(A, \cdot)$ be an abelian semigroup with $|A| \geq 2$ such that there exists $a, b \in A$ for which $a^2x^3b^2 = x^2, \forall x \in A$. Prove that $x^3 = x^2, \forall x \in A$.
I first tried ...
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0
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Prove that a semigroup $S$ in which $x^3=x$ and $x^2y^2=y^2x^2$ for every $x,y \in S$ is commutative. [duplicate]
Prove that a semigroup $S$ in which $x^3=x$ and $x^2y^2=y^2x^2$ for every $x,y \in S$ is commutative.
My attempt: From $x^3=x$, I can deduct that $x^n=x$ if $n$ is odd and $x^n=x^2$ if $n$ is even for ...
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7
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0
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162
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Proving a semigroup is abelian [duplicate]
Let $M$ be a set with an internal binary operation, denoted $\cdot$. Knowing that $(M, \cdot)$ is a semigroup and if there exists a natural number $n$ such that $x^ny^n=yx$, $\forall (x, y) \in M^2$, ...
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1
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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 $$x^2y^2=(x^2y^2)^2\quad\text{and}\...
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3
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1
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When does this construction always yield a congruence?
Suppose $\mathcal{M}$ is a commutative monoid and $E$ is any equivalence relation on $\mathcal{M}$. Define $\widehat{E}$ to be the "shift-invariant" part of $E$, that is, $$a\widehat{E}b\...
- 229k
0
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0
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36
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Lipschitzness of product map on $L^2 \times L^2$
Is the map $(u,v) \rightarrow uv$ a Lipschitz map from $L^2(\Omega) \times L^2(\Omega) \rightarrow L^2(\Omega)$ for $\Omega \subset \mathbb{R}^n$?
I am particularly interested in existence of local ...
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0
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50
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Orbit-Stabilizer Theorem for Semi-groups?
Given a point $x \in X$ the set of group G elements $$ G^{x} = \{ g \in G: g.x = x\}$$ is called the stabilizer group of x.
Orbit Stabilizer Theorem for Groups: Each left coset of $G^x$ in G is in one-...
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6
votes
1
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207
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When does a semigroup homomorphism preserve identities on monoids?
Let $X,Y$ be monoids, with identities $e_X,e_Y$, respectively. Let $f:X\to Y$ be a semigroup homomorphism. That is, any function which satisfies
$$f(xy)=f(x)f(y)\quad\forall x,y \in X\tag{1}$$
I know ...
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0
votes
1
answer
34
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Ones in subsemigroups
I found the following definition in a PDF.
Definition 2.1. Let S be a semigroup and ∅ 6= T ⊆ S. Then T is a subsemigroup of S if
a, b ∈ T ⇒ ab ∈ T. If S is a monoid then T is a submonoid of S if T is ...
2
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0
answers
21
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A nice characterization of commuting idempotent endomaps?
Let $X$ be a set, and let $f\colon X\to X$ be a map. It is idempotent when for each $x\in X$, $f(f(x))=f(x)$. It is equivalent to the requirement that the restriction of $f$ on its image $f(X)$ is the ...
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0
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0
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26
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How to find for which of the given possible relations there is bijectivity from a set PG to the semigroup G on GAP?
Let G be abelian semigroup generated by 1,x,y and z with the relations:
$1^2=1,x*1=x,1*y=y,1*z=z,x^{290}=x^0=1,y^{290}=y^0=1,x^{290}=z^{0}=1,$
$x^{A}=y^{U}*z^{P},y^{V}=x^{B}*z^{Q},z^{R}=x^{C}*y^{W}.$
...
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0
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0
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42
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Is there even a closed-form formula for the number of associative operations on an $n$-element set?
I have read many questions that ask for a closed-form formula for the function $f$ that takes a natural number $n$ and outputs the number of associative operations on an $n$-element set. So far, we ...
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3
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0
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25
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Domain of the infinitesimal generator of subordinators
Let $(X_t)$ be a subordinator (not killed). Since $(X_t)$ is a non-decreasing Levy process, we have the corresponding infinitesimal generator:
\begin{equation}
Af(x)=\delta f'(x)+\int_{0}^{\infty}(f(x+...
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2
votes
1
answer
111
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Is $eSe \cong S^1$ possible?
Let $S$ be a semigroup. Denote $S^1 := S\sqcup \{1\}$ (adding external identity) - so $1x=x=x1$ for every $x\in S$. In present case we add the external identity whether or not $S$ is a monoid.
For ...
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1
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0
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37
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Rees factor semigroup is the zero semigroup
I've been reading up some semigroup theory and have come across the concept that Rees factor semigroups are the zero semigroups and was wondering how to prove that $S/L$ is the right zero semigroup, ...
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0
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0
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39
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How to find which of the given components satisfy the existence of a certain bijectivity on GAP
I'm trying solve the following problem:
Let $Q$ be a given set and $G$ the monoid generated by $x,y,z$ nonnegative integers.
How to find which of the given components of $Q$ satisfies the condition ...
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1
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0
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30
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Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
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0
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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 $S=\cup_{x\...
- 367
0
votes
1
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46
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The notion of (uniformly) bounded analytic semigroups
From the book "$C_0$-semigroups and Applications" of Vrabie, we have this defintion of uniformly bounded analytic semigroups:
Definition 7.1.1. Let $\mathbb{C}_\theta=\{z \in \mathbb{C} ;-\...
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1
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1
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Let $S$ be a semigroup. If for any $x,y\in S$, $x^2y=y=yx^2$, then prove that $S$ is an abelian group
Let $S$ be a semigroup. If for any $x,y\in S$, $x^2y=y=yx^2$, then prove that $S$ is an abelian group.
My solution goes like this:
If for any $x,y\in S$, we have $x^2y=y=yx^2$. Then this implies $x^2=...
- 1,805
2
votes
1
answer
40
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Proving a semigroup properties are equivalent
I want to prove that the two clauses for a semigroup being a group are equivalent but am not sure how to go about it?
Prove the following are equivalent
$$1)\quad\forall a\in S, aS=S \quad \text{and} ...
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1
vote
0
answers
19
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Commutator estimates between a contraction evolution group and a differential operator
In the Appendix of "Commutator Estimates and the Euler and Navier-Stokes equations", Kato proves the following inequality
\begin{equation}\tag{1}\label{1}
\|J^s(fg)-f(J^sg)\|_{L^p}\lesssim \|...
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0
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13
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Find a compact set $S$ in topological semigroup $T$ with $SK\subseteq (T-K)$ for some compact set $K$
Let $T$ be a topological semigroup and $K$ be a compact set in $T$. I know that if $T=(\mathbb{R}^n, +)$ or $T=(\mathbb{Z}^n, +)$, then for every compact set $K$ in $T$, there is compact set $S$ with $...
- 1,157
0
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0
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19
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Compactification of a topological semigroup with right-identity structure
I'm looking for a hint on this question:
Let $T$ be as a semigroup with right-identity structure i.e. $rs=r$ for all $r, s\in T$. Also, we consider $T$ as a topological semigroup that is a locally ...
- 1,157
4
votes
2
answers
64
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Regular semigroups -- intuition
I'm trying to develop some intuition around the definition of the pseudoinverse in a regular semigroup.
Let the semigroup be $S$ with its associative operation written by juxtaposition. The ...
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votes
1
answer
36
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A semigroup with right identity and right inverses is a group.
Let $G$ be a nonempty set closed under an associative product, which in addition satisfies:
(a) There exists an $e\in G$ such that $a· e = a$ for all $a\in G$.
(b) Give $a\in G$, there exists an ...
- 1,805
1
vote
1
answer
55
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Infimums of subsets of monoid.
I came up with the following idea This may have already been studied by someone else or may be a common sense idea in this field.
Let $(M,+,\leq)$ be a totally ordered commutative monoid. For any non-...
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3
answers
104
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$(ab)^2=a^2b^2$ doesn't hold for semigroup.
I am trying to prove that
$(ab)^2=a^2b^2$ doesn't hold for a semigroup $S$. It is easy to prove(using inverse property) that the above condition holds for an abelian group. However, for semigroup, it ...
- 1,272
0
votes
0
answers
36
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How to prove this is a Group? [duplicate]
On class, my teacher said that,if set $G$ is equipped with an operation: $G\times G\to G:(a,b)\to ab$ that satisfies:
(i) $(ab)c=a(bc)$ $\forall a,b,c\in G$
(ii) $\exists e\in G$ s.t. $ae =a$ $\forall ...
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0
answers
70
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Do some authors call a "monoid" a "semigroup"?
I'm reading A Short Course on Operator Semigroups by Engel and Nagel, and they say on page 2:
A family $\left(T(t)\right)_{t \geq 0}$ of bounded linear operators on a Banach space is called a ...
- 2,297
2
votes
1
answer
63
views
Cancellation property of the Stone Cech compactification
Let $G$ be a discrete countable group and let $\beta G$ be the Stone-Cech compactification of $G$, which has the structure of a semigroup.
Is $\beta G$ left cancellable? What about right cancellable? (...
1
vote
1
answer
228
views
Krohn-Rhodes decomposition for transformations over $\{0,1\}$
I'm trying to learn about the Krohn-Rhodes theorem, and I'm struggling to apply it even on incredibly simple semigroups.
Notation
$C_2 = \langle e,x \mid x^2=e \rangle$ is the cyclic group on two ...
- 121
4
votes
0
answers
45
views
Counting Binary Sequences Arising from a Numerical Semigroup
This is a sequel to this question, though none of the background there should be necessary. Yly's excellent answer to that question gives a few different ways of stating the below question for $n = 2$,...
- 665
12
votes
3
answers
388
views
Is there a minimal generating set of reals which additively generate all the reals?
Is there a set $S$ of real numbers such that the submagma generated by $S$ under addition is the entire set of real numbers, but such that no proper subset of $S$ generates the entire set of real ...
- 16.8k
8
votes
0
answers
154
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Which ultrafilters are (sort-of) commutators?
There is a natural way to extend a binary operation $\star$ on $\mathbb{Z}$ to a semicontinuous binary operation $\widehat{\star}$ on $\beta\mathbb{Z}$, the latter being the set of ultrafilters on $\...
- 229k
4
votes
1
answer
65
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Are there theorems about extending a monoid without inverses to a group by reflecting the elements? [duplicate]
What I have in mind is something like the following: the natural numbers with addition form a monoid. You can imagine constructing the integers by taking the naturals, adding a set constructed by &...
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1
vote
1
answer
92
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Is there a monoid with universal property dual to that of a free monoid?
One of the important properties of a free monoid on a set is that for any other monoid, you can create a unique homomorphism from the free monoid to that monoid.
Does there exist some monoid over a ...
- 11
1
vote
3
answers
92
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Let $e$ be an idempotent of the monoid $M$, $x$, $y$ be two elements of $eMe$. Then, $(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$ if and only if $MxM = MyM$ [duplicate]
I am studying semigroups. I saw a Lemma in the text that states:
Let $e$ be an idempotent of the monoid $M$, $x$, $y$ be two elements of $eMe$. Then, $(eMe)\,x\,(eMe) = (eMe)\,y\,(eMe)$ if and only ...
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