All Questions
1,123 questions
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Separable nonassociative algebras
In his paper "Structure and Representation of Nonassociative Algebras", Schafer notes that an arbitrary nonassociative algebra over a field is separable "if and only if the ...
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0
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105
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Flag variety as monoid and Schubert calculus
The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation.
When looking ...
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Conjugacy classes of monoids II: Abelianising a monoid, wrongly
$\newcommand{\unsim}{\mathord{\sim}}$Let $G$ be a group. What is
$$
G/\left(ab\sim ba\ \middle|\ a,b\in G\right)?
$$
Answer: not $G^{\mathrm{ab}}$, but the set of conjugacy classes of $G$.
When ...
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63
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Maximal orders separable over their centre
Let $\mathcal{A}$ be a central simple $K$-algebra, where $K$ is an algebraic number field. It is known that $\mathcal{A}$ is separable over $K$ (following the definition of DeMeyer and Ingraham's book)...
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2
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270
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Road map: beyond Artin-Wedderburn theorem
For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...
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586
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Brauer group of $\mathbb{Z}_{(p)}$
This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
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83
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What is known about the algebraic completion of a monoid?
It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid:
Let $W$ be a monoid and let $p(x)=q(...
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2
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409
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Extending monoids to a ring
I started reading about monoids (and semigroups in general) and came across of the example of some non-commutative monoids which cannot be endowed with some addition turning it into a ring (the monoid ...
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Has an "algebraic manifold" been defined before? Are there any non-trivial examples?
Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-closed if the following condition holds:
for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $...
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250
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Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
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805
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Cancellation property for commutative monoid
Let $(M,+,e)$ be a commutative monoid with unit $e$. An element $a\in M$ is called cancellative element if
for any $b,c \in M$ such that $a+b=a+c$ implies that $b=c$.
Let $(\mathbf{N},+,0)$ the ...
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What is the real name for the initial object in the category of "monoid-valued measures of intervals" on transitive relations?
(I'm not asking for a true/false answer; I have a true algebraic fact and I'm looking for a reference in the literature. By the way, there is a version of this theorem that replaces monoid with $R$-...
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52
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Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?
The terms are defined in a related question. [1]
Conjecture 1. Let $A$ be a set, $W$ a cancellative invertible-free monoid, and $\cdot\colon A\times W\rightarrow A$ a cyclic right $W$-action generated ...
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Is every invertible-free cancellative monoid action represented by "shifting" certain maps?
[Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments]
Let $W,X$ be ...
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284
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Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
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terminology for a kind of two-sided module over a monoid
If $M$ is a monoid object in a pointed category $\mathcal{C}$, then a right $M$-module is an object $X$ equipped with a morphism $\alpha: X\times M\to X$ that satisfies the usual rules. There are ...
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422
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Generalized cancelation properties ensuring a monoid embeds into a group
Context: an obvious necessary condition for a monoid to embed into a group (as submonoid) is to satisfy the left and right cancelation rules:
$$xy=xz \quad\Longrightarrow y=z;$$
$$yx=zx \quad\...
- 63.9k
5
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Can every cancellative invertible-free monoid be embedded in a group?
A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product ...
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161
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Are cofibrations in topological monoids preserved by forming the product with the identity?
Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...
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The type number of an algebra
I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
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499
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Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
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249
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"Completion property" in $(\beta\omega,+)$
Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, ...
- 48.9k
4
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222
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Addition and Rudin-Keisler ordering in $\beta \omega$
$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends ...
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355
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On logarithmic schemes
I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
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164
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non-abelian tensor products of several groups
R. Brown and J-L. Loday had defined the tensor product of two arbitrary groups acting on each other. Let $G,H$ be groups with actions on each other on the right. each group act on itself by ...
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132
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Integral monoid rings and Ore conditions
Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.
I have two, ...
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197
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Top and bottom composition factors of $M$ are isomorphic
Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...
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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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Color algebras and color involutions
If $A$ is a $G$-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined ...
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43
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Can Q(R) embed to Q((R ⊗ S )/ P)
Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian.
let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...
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193
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$R/I\cong R/\text{Ann}_R(R/I)$ but $I\neq\text{Ann}_R(R/I)$
I originally asked this on Stack Exchange but with no luck. Upon doing research with some noncommutative rings, I thought of a curious question. Does there exist a noncommutative unital ring $R$ and ...
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Heuristics for the word problem for monoids
The question is about a purely practical problem:
Given is a list of identities, as in http://www.findstat.org/MapsDatabase/Mp00069:
...
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41
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Characterizing centralizer of nilpotent self-maps
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(...
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Extending a monoid object in a category
A monoid object in a pointed category $\mathcal{C}$ is an object $M$ equipped
with a multiplication morphism $\mu: M\times M\to M$ that is associative and unital, meaning that the diagrams that ...
- 12.5k
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96
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Gelfand–Kirillov dimension of the first Weyl algebra by using the definition
$\DeclareMathOperator\GKdim{GKdim}$Here I am trying to find the Gelfand–Kirillov dimension of the first Weyl algebra just by using the definition of the Gelfand–Kirillov dimension.
Let $A$ be an ...
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163
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Classifying spaces of amalgamated topological monoids
Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
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142
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On the width of the Catalan monoid and the rank of K-groups of the Furstenberg transformation group
The semigroup algebra of the Catalan monoid is isomorphic to the incidence algebra of $P_n$, where $P_n$ is the poset consisting of subsets of { 1,...,n } where for two subsets $X \leq Y$ if and only ...
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409
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Does every set have a rigid self-map?
The question was asked on Mathematics Stackexchange
but has remained unanswered so far.
A self-map is a map $f:X\to X$ from a set $X$ to itself. There is an obvious notion of morphism, and thus of ...
- 4,416
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89
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Semigroups associated to binary necklaces and their semigroup algebra
I came across the following semi-group and the associated finite dimensional semi-group algebras over a field $K$ (which are Nakayama algebras) as they have very nice homological properties. My ...
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5
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"Tietze-like transformations" for defining interesting bijections between algebraic structures
Consider the following two definitions of the natural numbers:
The natural numbers are the algebraic structure $\mathbb{N}_1$ generated by one constant, $0$ and one unary function, $S$ (and no ...
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498
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Weyl algebra as an Azumaya algebra over its centre
Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
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241
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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,...
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2
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449
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Survey of recent developments of the Gelfand-Kirillov dimension
It is almost two decades since the now classical books by McConnell and Robinson's
[ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...
- 3,318
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82
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On noncommutative transcendence degrees
The original transcendence degree for (noncommutative) division algebras is the Gelfand-Kirillov transcendence degree, due to I. Gelfand and K. Kirillov ([ Sur les corps li´es aux algèbres ...
- 3,318
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3
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908
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Does Morita theory hint higher modules for noncommutative ring?
Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. ...
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244
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On the Gelfand-Kirillov Conjecture
The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
- 3,318
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219
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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
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3
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non-associative but commutative algebra [closed]
Is it possible(or may be easier) to give an example of non associative algebra but commutative?
At first sight, it seems possible to prove associativity from commutativity but later realised it may no ...
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246
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Surjective monoid homomorphism $\text{End}(B)\to \text{End}(A)$ given surjection $g:B\to A$
For any set $A\neq\varnothing$ let $\text{End}(A)$ denote the endomorphism monoid, consisting of all functions $f:A\to A$, together with composition. If $A, B\neq \varnothing$ are sets and $g:B\to A$ ...
- 48.9k
3
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1
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197
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Gelfand-Kirillov dimension of the first Weyl algebra
How can we compute the Gelfand-Kirillov dimension (GK for short) of the first Weyl algebra?
As we know we can look at the Weyl algebra as a generalized Weyl algebra in the following way:
Let $A=\...
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