All Questions
1,123 questions
8
votes
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322
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Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
8
votes
1
answer
645
views
Isomorphic morphisms. A 27-morphism category
Two morphisms of category $\ \mathbf C\ $ are isomorphic to one
another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that ...
- 4,786
8
votes
1
answer
445
views
For $G=\mathbb{Z}^2\rtimes \mathbb{Z}$, $Spec(\mathbb{Z}G)$=?
Let $G$ be the group $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL_2(\mathbb{Z})$, which ...
- 1,528
8
votes
2
answers
262
views
An operation on binary strings
Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ by $\alpha$ and each ...
- 9,720
8
votes
2
answers
352
views
Non-density of invertible elements in $\ell_1(\mathbb{N}_0)$
Consider the Banach algebra $\ell_1(\mathbb{N}_0)$ (with convolution / Cauchy product of series). I am looking for an elementary proof of the fact that the group of invertible elements in this algebra ...
- 11.3k
8
votes
1
answer
236
views
Quiver and relations for a monoid related to Catalan numbers
Let $C_n$ be the monoid consisting of monotone maps $\{1,...,n\} \rightarrow \{1,...,n\}$ with $f(i) \leq i$ for all $i$.
The cardinality of $C_n$ is given by the Catalan numbers.
Consider $A_n= \...
- 26.5k
8
votes
1
answer
314
views
What algebras are quotients of $\ell_1(\mathbf{N})$?
Every separable Banach space is a linear quotient of $\ell_1$, however not every separable Banach algebra is a Banach-algebra quotient of $\ell_1(G)$ for some group $G$ (these are the so called ...
- 331
8
votes
2
answers
483
views
Posets obtained from a semigroup by the definition $x \leq y \iff x \cdot y = x$
A po-groupoid is a groupoid $\langle A,\cdot\rangle $ such that the relation
defined by
$$
x \leq y \text{ if and only if } x \cdot y = x
$$
is a partial order on $A$, the order related to $\langle ...
- 1,124
8
votes
1
answer
319
views
Über theorem on unavoidable patterns?
Let $A$ be an alphabet of $k$ symbols,
and $p$ a pattern.
An example of a pattern is $p=XX$, where $X$ is any finite
string of symbols from $A^+$.
Avoiding $p$ is avoiding any subword repeated twice ...
- 151k
8
votes
1
answer
623
views
Is there a good computer package for working with complexes over non-commutative rings?
I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...
- 44.7k
8
votes
1
answer
437
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
8
votes
2
answers
425
views
Dimension of commutative subalgebras of a central simple algebra
let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division $k$-...
- 81
8
votes
1
answer
248
views
Are annihilation modules in the quantum torus necessarily principal?
I hope that my question yields some standard fact from (noncommutative) ring theory. In discussions with other graduate students, we have outlined some approaches to tackling the question, but haven'...
- 54.6k
8
votes
2
answers
826
views
Which semigroups can be linearly ordered?
As usual I consider a semigroup to be a structure $(A, +)$ such that $+$ is an associative binary function over the set $A$. The notion of linearly-ordered semigroup corresponds to structures of the ...
- 768
8
votes
1
answer
922
views
What's the cokernel of a monoid homomorphism?
Let $f:A\to B$ be a monoid homomorphism. Where can I find an explicit description of the its cokernel? Are there any books on this topic?
By the cokernel of $f$, I mean the universal arrow which ...
- 10.5k
8
votes
1
answer
238
views
Functions over monoids which factor in two different ways
This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
- 32.1k
8
votes
2
answers
708
views
Quantum Grassmannians?
In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For ...
- 449
8
votes
1
answer
261
views
A generalisation of $C_0$-semigroups
A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an ...
- 3,388
8
votes
1
answer
1k
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Lattice-ordered commutative monoids
By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...
- 44.4k
8
votes
2
answers
1k
views
Sets of matrices which are irreducible but not strongly irreducible
A set of $d \times d$ real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger ...
- 6,206
8
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1
answer
229
views
Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups
An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
- 10.5k
8
votes
1
answer
451
views
Separable and finitely generated projective but not Frobenius?
Let R be a commutative ring, and $A$ an $R$-algebra (possibly non-commutative). Then $A$ is separable if it is finitely generated (f.g.) projective as an $(A \otimes_R A^{\mathrm{op}})$-algebra. ...
- 27.5k
8
votes
1
answer
352
views
Constructive definition of noncommutative rational functions (aka free skew fields)
The question
Let $F$ be a field. (I am fine with assuming $F=\mathbb{Q}$, but I suspect
that a "right" answer will be independent of $F$.) Let $k$ be a nonnegative integer.
Question. Is ...
- 33.8k
8
votes
0
answers
411
views
Semigroups of matrices closed under conjugate transposition
An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
- 563
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
8
votes
0
answers
354
views
Rough paths, unparametrized path space, and Kontsevich's moduli space of stable maps
Let $X$ be a manifold. Modulo reparametrization, the path space of $X$ is a groupoid $\Pi_X$. In Kapranov's "Free Lie Algebroids and the Space of Paths", Kapranov constructs an associated ...
- 31
8
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0
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219
views
Differential birational equivalence
Suppose the base field algebraically closed and of zero characteristic.
There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion ...
- 3,318
8
votes
0
answers
419
views
Are most semigroups nilpotent of degree 3?
A semigroup is nilpotent of degree 3 if every product of 3 elements gives the same result. In 2012, Andreas Distler and James D. Mitchell wrote that:
It is part of the folklore of semigroup theory ...
- 22.3k
8
votes
0
answers
270
views
Does this kind of non-noetherian bimodule exist?
Question: Do there exist simple rings $R$ and $S$ (i.e., rings with no proper nonzero ideals) and an $(R,S)$-bimodule $M$ such that
$M$ is finitely generated both as a left $R$-module and a right
$...
- 5,407
7
votes
3
answers
2k
views
Units in a group algebra
Let k be a field and let G be a finite group. I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, ...
- 575
7
votes
4
answers
1k
views
Torsors for monoids
Torsors are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful.
In general I'm interesting in the ...
- 1,882
7
votes
2
answers
587
views
Representation theory of the full linear monoid
The full linear monoid $M_N(k)$ of a field $k$ is the set of $N \times N$ matrices with entries in $k$, made into a monoid with matrix multiplication.
A representation of $M_N(k)$ on a vector space $V$...
- 22.3k
7
votes
2
answers
370
views
Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.
First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
- 4,842
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
7
votes
2
answers
489
views
How big can a commutative subalgebra of Weyl algebra be?
Consider the smallest Weyl algebra $A_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I ...
- 605
7
votes
2
answers
383
views
Counting nilpotent self-maps of $\{1,\dots,n\}$ with image of a given cardinal
Let $\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(...
- 355
7
votes
2
answers
1k
views
Is there a notion of point in noncommutative geometry?
It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
- 136
7
votes
2
answers
544
views
A linearly orderable monoid which does not embed into a linearly orderable group
It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable ...
- 10.5k
7
votes
1
answer
266
views
Positive cone of a subgroup of $\mathbb{Z}^n$
This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G_+=G\cap\mathbb{Z}_{\ge0}^n$ be a ...
- 16.9k
7
votes
1
answer
281
views
Question concerning the coefficients of block idempotents
Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
- 1,755
7
votes
2
answers
284
views
Making a non-unital algebra the unique maximal one-sided ideal in a unital algebra
If $R$ is an algebra without a unit, then the standard unitisation $R^\sharp$ can have maximal one-sided ideals other than $R$. Thus, it is natural to ask about the following. Let $R$ be an algebra ...
- 71
7
votes
1
answer
653
views
Which CAS can do basic non-commutative differential algebra?
This is a repost of my question at MSE from 7 months ago, to which I haven't been able to find an answer yet.
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute ...
- 7,127
7
votes
1
answer
361
views
For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group
For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...
- 798
7
votes
1
answer
457
views
Are perfect complexes the same as compact objects in D(R) for noncommutative rings?
The Stacks Project proves Thomason's insight that
compact objects of the derived category $\simeq$ bounded complexes of finitely generated projective modules
in Section 15.78, but the running ...
- 71
7
votes
1
answer
179
views
Symmetry of unique generator property
In this article:
Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503.
the author defines a ring $R$ to ...
- 1,507
7
votes
1
answer
271
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 413
7
votes
1
answer
722
views
How is called a semigroup...
Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
- 3,110
7
votes
1
answer
650
views
Cones, monoids, and the space of (very) ample divisors
An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
- 1,871
7
votes
1
answer
444
views
Which monoids have a faithful irreducible representation?
Let $*$ be a binary operation on a set $M$, with an identity element $e\in M$.
A monoid representation of $(M,*,e)$ is a map $\delta:M\to (S\to S)$ for some set $S$, such that $\delta(e)=\mathrm{id}_S$...
- 24.8k
7
votes
1
answer
241
views
Lie monoids as monoids internal to the category of smooth manifolds?
This question can be thought as a complement to this one.
Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups,...
- 321