All Questions
393 questions with no upvoted or accepted answers
33
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0
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2k
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Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
- 5,309
21
votes
0
answers
869
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
- 28.6k
17
votes
0
answers
536
views
Question about combinatorics on words
Let $\{a_1,a_2,...,a_n\}$ be an alphabet and let $\{u_1,...,u_n\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$...
user6976
16
votes
0
answers
861
views
Is "being a full ring of quotients" a Morita invariant property?
Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
- 2,454
15
votes
0
answers
1k
views
Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?
This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...
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13
votes
0
answers
251
views
Is every simply connected finite complex the classifying space of a finite monoid
On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
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12
votes
0
answers
543
views
Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
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12
votes
0
answers
321
views
Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
12
votes
0
answers
185
views
Hopf-Galois extensions where the "extension" is a module?
For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...
- 10.4k
12
votes
0
answers
267
views
Finitely generated skew-fields
There is a well known theorem saying that a commutative field that is finitely generated as a ring has to be finite (Kaplansky).
Is the same true for non-commutative "fields" (usually called ...
12
votes
0
answers
533
views
Does there exist a Noetherian ring of finite injective dimension but higher Krull dimension?
Definition: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if
(i) $R$ has finite left and right injective dimension (in which case it turns out ...
- 221
12
votes
0
answers
443
views
Nullstellensatz for quaternionic plane curves?
By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after ...
- 17.6k
11
votes
0
answers
427
views
Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 631
11
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0
answers
286
views
Does every finite poset have a rigid endomorphism?
Crossposted on Mathematics.
In this post, an order-preserving self-map of a poset $X$ will be called an endomorphism of $X$, and such an endomorphism $f$ will be called rigid if the only automorphism ...
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11
votes
0
answers
265
views
Criteria for a map of rings to induce an equivalence on K-theory?
Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...
- 331
11
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0
answers
214
views
Is it decidable if a tree-presented semigroup contains an idempotent?
A semigroup presentation $\langle A | R\rangle$ is called tree-like if every relation has the form $ab=c$, $a,b,c$ are in $A$ and if two relations $ab=c, a'b'=c'$ belong to $R$, then $c=c'$ if and ...
user6976
10
votes
0
answers
248
views
What is the tiling semigroup for an einstein "hat" tiling?
My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
- 379
10
votes
0
answers
367
views
A formula for Frobenius number of certain numerical semigroups
The old formula for the Frobenius number of a numerical semigroup generated by two elements can be stated as follows: assume $\gcd\{a+1,b+1\}=1$, then the Frobenius number of $S= \left<a+1,b+1\...
- 30.5k
10
votes
0
answers
314
views
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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10
votes
0
answers
2k
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Is my definition of a context algebra new?
In my DPhil thesis, I defined what I called a context algebra as a model of meaning in natural language. The idea is to mathematically formalise the notion that meaning is determined by context. It ...
- 223
9
votes
0
answers
164
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Parallelizability of Lie monoids
A Lie monoid is a monoid, together with a structure of a smooth manifold (possibly with a boundary), such that the monoid multiplication is smooth.
If all left (or right) translations in a Lie monoid $...
- 61
9
votes
0
answers
347
views
What is the precise connection between logarithmic algebraic geometry and the field with one element?
Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
- 11.8k
9
votes
0
answers
273
views
About an algebraic construction of a sheaf of formal microdifferential operators
While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in ...
- 7,789
9
votes
0
answers
373
views
Embedding $\beta\mathbb{N}$ into a product of Cantor sets
Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
- 297
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
votes
0
answers
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
0
answers
227
views
On the structure of an algebra as a bimodule
$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
- 541
7
votes
0
answers
295
views
A minimal semigroup generating subset of the additive reals
I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
- 2,023
7
votes
0
answers
138
views
The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
- 41.9k
7
votes
0
answers
221
views
Strange formula for the dimension of a certain space of noncommutative polynomials
Consider a vector space $V_r(n)$ spanned by (noncommutative) monomials in variables $x_1,\ldots,x_r$
$$
x_{1}^{n_1}x_{2}^{n_2}\ldots x_{r}^{n_r}
$$
of total degree $n.$ Inside this space consider a ...
- 4,316
7
votes
0
answers
194
views
Factoring a function from a finite set to itself
Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
- 695
7
votes
0
answers
260
views
Generating the monoid of injective endomorphisms of the free group
Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...
- 121
7
votes
0
answers
579
views
Guises of the noncrossing partitions (NCPs)
From "Noncrossing partitions in surprising locations" by Jon McCammond:
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
- 10.5k
7
votes
0
answers
540
views
Algebraic-closures of division rings
In what follows, $x$ is always taken to commute with the coefficient ring. This means that for any given polynomial, you can put the coefficients to the right or the left of $x$ as you please. This ...
- 1,270
7
votes
0
answers
291
views
Differentially closed fields
Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$.
Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
- 2,751
7
votes
0
answers
438
views
How to prove that a projective module is not free?
Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...
- 777
6
votes
0
answers
79
views
Examples of $\mathbb{N}$-graded algebras whose global dimension is strictly less than the GK dimension
The relationship between the global and GK dimensions of Artin-Schelter regular algebras remains to be mysterious, yet both dimensions are conjectured to be equal.
In a more broad setting, are there ...
- 61
6
votes
0
answers
151
views
On dual notions of morphisms of algebraic structures obtained by replacing equaliser with coequalisers
This question is based on this discussion from the Category Theory Zulip. See also the earlier question Natural cotransformations and "dual" co/limits.
Let $G$ and $H$ be groups. We define ...
- 11.8k
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
votes
0
answers
259
views
Usefulness of total algebras and exotic generating series
In his first Algebra volume, Bourbaki [1] defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
- 3,738
6
votes
0
answers
183
views
Examples of groups with a positive homogeneous presentation without the Haagerup property or not of type $F_\infty$
I am looking for groups with a certain presentation that do not have the Haagerup property or are finitely presented but not of type $F_\infty$ (meaninig that for some $n\geq 3$ we cannot find any ...
- 61
6
votes
0
answers
177
views
Is the monoid of all cancellative finitely generated commutative monoids cancellative?
$\DeclareMathOperator\Mon{Mon}\DeclareMathOperator\Grp{Grp}$Let $\Mon'$ be the set of isomorphism classes of (small) commutative, unital, cancellative ($a + t = b + t$ implies $a = b$) monoids. It is ...
- 1,094
6
votes
0
answers
190
views
The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 41.9k
6
votes
0
answers
255
views
Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra
This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
- 33.8k
6
votes
0
answers
584
views
What are the topics in noncommutative algebraic geometry?
Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...
6
votes
0
answers
230
views
Gelfand ring in Bourbaki's exercises
In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that
The set $A^*$ ($=A^{-1}$) of invertibles is open.
The uniform structure ...
- 3,738