All Questions
1,123 questions
3
votes
1
answer
389
views
Dimension of hermitian rank at most $k$ matrices over quaternions
In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
- 3,209
2
votes
1
answer
176
views
Generating totally ordered free commutative monoids
Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$.
When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am ...
- 341
1
vote
1
answer
70
views
Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations
It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...
- 2,506
1
vote
1
answer
126
views
Do these sorts of submonoids go by a particular name?
Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows:
$$r(x)=\{y\in M:xy=x\}$$
$$l(x)=\{y\in M:yx=x\}$$
Do these sorts of sub-monoids go by a particular name?...
- 111
2
votes
0
answers
92
views
Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
- 527
10
votes
1
answer
673
views
Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?
Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
- 2,506
1
vote
0
answers
60
views
A variation on Dixmier's counterexample concerning centralizers in $A_1$
This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some ...
- 2,837
3
votes
1
answer
175
views
If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?
I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...
- 16.9k
8
votes
1
answer
448
views
Can a Shelah semigroup be commutative?
A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$...
- 41.9k
3
votes
1
answer
128
views
Size of a minimum generating set for full transformation monoids
Given any finite set $X$ the set $\mathcal{T}(X)=X^X$ of all functions from $X$ to $X$ clearly forms a monoid under composition. Now if we call any family of functions $\mathcal{F}\subseteq \mathcal{T}...
- 2,506
5
votes
0
answers
86
views
von Neumann regular ring homomorphisms
Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.
In particular, $\mathrm{id}...
- 858
4
votes
1
answer
158
views
Dimension of a module over a left-Ore domain
If $R$ is a domain, and $M$ a (left) $R$-module, what are the different notions of dimension of $M$ and their respective assets, what do they measure?
I found out that if $\dim_RM$ is the cardinal of ...
- 1,555
6
votes
1
answer
444
views
Homotopy type of a specific discrete monoid
Consider the discrete monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$, since $z$ is ...
- 3,901
5
votes
0
answers
219
views
Constructing a noncommutative algebra from a commutative algebra
I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
- 111
9
votes
1
answer
211
views
Reference for Kakutani result on power sum bases of symmetric functions
Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
- 528
2
votes
1
answer
244
views
radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices
Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set
$$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
- 1,209
5
votes
2
answers
332
views
Questions on weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called weakly symmetric in case $soc(P)=top(P)$ for every indecomposable projective module $P$ and it is called symmetric in case $D(A) \cong A$ as ...
- 26.5k
1
vote
0
answers
40
views
Relation between left projections
Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$.
Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...
- 4,058
3
votes
1
answer
122
views
Partition theorems for located words
In this paper Bergelson, Blass, and Hindman prove the following
Theorem 1.2 Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. ...
0
votes
1
answer
655
views
How to show two semigroups are isomorphic?
I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the ...
- 233
1
vote
0
answers
38
views
The statue of a sequence of finite projections
Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false?
Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...
- 4,058
1
vote
0
answers
38
views
something concerning finite projections
Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.
Q. Can we say that ...
- 4,058
13
votes
2
answers
317
views
Semigroup of differentiable functions on real line
Let $D(\mathbb R) $ be the set of all differentiable functions $f: \mathbb R \to \mathbb R$. Then obviously $D(\mathbb R)$ forms a semigroup under usual function composition. Can we characterize (up ...
- 235
1
vote
2
answers
126
views
Upper triangular $2\times2$-matrices over a Baer *-ring
Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices
$\left(\begin{array}{cc}
a_1& a_2 \\
0 & a_4
\end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...
- 4,058
1
vote
0
answers
116
views
A generalized Cauchy type functional equation
Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...
- 1,209
0
votes
0
answers
213
views
make me idempotent
$T_n$ be the full transformation semigroup on $X_n= \{1, 2, \cdots , n\}$.
$D_r =\{\alpha \in T_n: |im(\alpha)|=r\}$.
$E(D_r)$ is the set of all idempotents of semigroup $T_n$.
$support(\alpha)=\{...
- 109
3
votes
1
answer
189
views
Faces of polyhedral cones and open immersions of affine toric schemes
Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.
Let $\sigma\subseteq V$...
- 6,700
6
votes
0
answers
92
views
What quantum groups admit quantum topography space structure?
Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
- 361
2
votes
1
answer
74
views
Strongly finite projections in $*$-rings
Let $A$ be a $*$-ring. Let us have some points:
i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.
ii) On the set of projections, we write $p\leq q$ if $pq=p$.
iii)...
- 4,058
2
votes
0
answers
68
views
Transmission of finite projections
Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$).
Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...
- 4,058
6
votes
0
answers
340
views
Would you like a subject class for semigroup theory on the arXiv?
After contacting the arxiv recently about possibly adding semigroup theory as a subject class, they suggested I canvas the research community to establish whether such a subject class would be used ...
- 77
6
votes
0
answers
117
views
Closedness of the partial order in complete Hausdorff semitopological semilattices
First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
- 41.9k
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
user111492
1
vote
1
answer
106
views
Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
6
votes
1
answer
414
views
Is every (left) graded-Noetherian graded ring (left) Noetherian?
I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...
- 953
10
votes
2
answers
716
views
On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
- 613
2
votes
0
answers
153
views
Algebraic version of unilateral shift
It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...
- 4,058
1
vote
0
answers
67
views
Free module over $H$-module algebra
Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(...
- 383
10
votes
1
answer
579
views
Group completion of topological monoids
Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\...
- 1,252
3
votes
1
answer
160
views
Differential operators and rules Ore polynomial
(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.)
When dealing with (nonlinear) dynamical systems, one often deals with state space ...
- 131
2
votes
1
answer
78
views
CAS implementing free algebras with involution
Is there any software that easily allows to make symbolic computations with involutions and homomorphisms? I need to define a product in an associative algebra with an (abstract) involution and ...
- 2,992
5
votes
0
answers
104
views
Finitely generated submodules of projectives lie inside f. g. projectives?
Let $R$ be a (not necessarily commutative) ring.
If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that
$M \subseteq P'...
- 51
1
vote
1
answer
110
views
Indecomposable monoids
Let $M$ be a commutative reduced and cancellative monoid and $K(M)$ its group of quotients.
We say that $M$ is indecomposable if for every divisor-closed submonoids $M_1$ and $M_2$, $M=M_1\oplus M_2$...
- 933
7
votes
1
answer
235
views
name for monoids inducing bimonoids in Rel?
Let Rel be the category of sets and relations, which is a (compact closed) symmetric monoidal category under the cartesian product of sets. We write $A \nrightarrow B$ to indicate a relation from $A$ ...
- 3,393
3
votes
0
answers
80
views
On the compactification of partial semigroups
We begin by introducing some relevant definitions.
Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...
- 745
3
votes
0
answers
169
views
Hochschild homology and Chern character quiver with potential
I am a beginner in quiver theory so this question might not be suitable for mathoverflow.
Let $(Q,W)$ be a quiver with potential and let $\Gamma$ be the Ginzburg DG-algebra associated to $(Q,W)$. Is ...
- 7,300
4
votes
1
answer
243
views
number of indecomposable summands of an extension of two modules
I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...
- 333
-4
votes
1
answer
224
views
Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? [closed]
Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there ...
- 48.9k
1
vote
0
answers
82
views
What is known about the cohomology of the matrix monoid?
When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object).
Let $M_n(R)$ be the monoid of matrices with matrix ...
- 1,726
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
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