All Questions
1,123 questions
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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
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367
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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
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314
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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 ...
- 1,914
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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 ...
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2
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2k
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A ring for which the category of left and right modules are distinct
What is an example of a ring $R$ for which the abelian category of left $R$-modules is not isomorphic to the category of right $R$-modules?
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4
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1k
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When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)
The title has it all. I'm looking for a reference to the following:
Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
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Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring
Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?
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2
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667
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Semi group of polynomials which all roots lie on the unit circle
Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $.
The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials.
With the standard multiplication, $X$...
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7
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2k
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Hochschild/cyclic homology of von Neumann algebras: useless?
Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...
- 5,425
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2
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1k
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Ternary associative multiplication
In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such ...
- 4,780
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2
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2k
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Global dimensions of non-commutative rings
This is related to my previous question: When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? I now would like to know the global dimension of the ring $R=\mathbb{C}\langle x_1,\dots,x_n\rangle/...
- 1,663
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3
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1k
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Structure Theorem for finitely generated commutative cancellative monoids?
Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...
- 63.1k
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4
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954
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Morita equivalence and moduli problems
Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules).
Ex: $M_n(R)$ (the algebra ...
9
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2
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2k
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What is the free monoidal category generated by a monoid?
In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category $...
- 1,446
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1
answer
679
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Mathematical software for computing in integral group rings of discrete groups?
I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...
- 2,758
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2
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945
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Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?
Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...
- 2,310
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2
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3k
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Jacobson radical = intersection of all maximal two-sided ideals
I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...
- 33.8k
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5
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1k
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References/literature for pushouts in category of commutative monoids? [ed. - amalgams]
This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid ...
- 25.8k
9
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1
answer
743
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Why is choice needed in Ellis' Lemma?
Ellis Lemma on idempotent elements asserts that:
Lemma (Ellis). Every compact semigroup has an idempotent.
The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.
...
- 1,422
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1
answer
248
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Categories of modules generated under coproducts by a small set?
Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$?
Equivalenty, for which rings $R$ does there ...
- 63.9k
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1
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211
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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
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1
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510
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Point modules of quantum projective space $\mathbb{P}^n$
Let $A$ be a quantum $\mathbb{P}^n$ defined by
$$
A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}.
$$
I would like to know the set $X$ of isomorphism ...
- 1,663
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1
answer
236
views
Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\...
- 985
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193
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Detecting/Characterising positive elements in free groups
Let $X$ be a set, and let $F(X)$ be the free group generated by $X$.
I will say that an element of $F(X)$ is positive if it is in the monoid generated by all the conjugates in $F(X)$ of every member ...
- 2,178
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1
answer
509
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Maximal localizations of von Neumann algebras
Suppose M is a von Neumann algebra.
Denote by L its maximal noncommutative localization,
i.e., the Ore localization with respect to the set of all left and right regular elements,
i.e., elements whose ...
- 37.8k
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1
answer
154
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Inductive and reducible functions
The question was asked by a Computer Scientist and is closely related to parallel computing. But it is clearly of algebraic nature, so I decided to post it here.
Let $X$ be a set and $\bar X$ be the ...
user6976
9
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1
answer
457
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Deformations of Ext rings
Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
- 4,133
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889
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Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 10.5k
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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 $...
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347
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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
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272
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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 ...
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373
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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 ...
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3
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1k
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Are all free monoids residually finite?
I cannot manage to prove that a free monoid with operation concatenation, and with at least two generators is residually finite. If there is just one generator, the free monoid $\{a\}^*$ is isomorphic ...
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2
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Reason to apply the Koszul sign rule everywhere in graded contexts
The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded-commutative algebra generated by $n$ elements of ...
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4
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1k
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Haar Measure on Locally Compact Semigroups
I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure?
...
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3
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609
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Nielsen-Schreier theorem for monoids
Let $S$ be a finitely generated free abelian semigroup (or monoid), and let $T \subset S$ be a sub-semigroup (sub-monoid). Does the Nielsen-Schreier theorem hold in this case, that is, will $S$ still ...
- 459
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2
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720
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Hochschild homology of upper triangular matrix algebra?
Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$?
Is there any ...
- 1,589
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Give an example of monoid with property $m^2 = m^3$
Give an example of finitely generated, infinite monoid $M$ with property that for all $m \in M$ we've got $m^2 = m^3$.
This question comes from the problem I was given during algebraic languages ...
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2
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Explicit description of a quaternion algebra with a prescribed set of ramified places
Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that $Q$...
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427
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Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF?
The title question is too vague so let me be specific.
Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ...
- 38.6k
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459
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Semisimple-ish rings!
Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not ...
- 279
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What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would ...
- 7,237
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3
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431
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Smallest faithful representation of an upper-triangular matrix quotient
This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...
- 33.8k
8
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539
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Vanishing of Hochschild homology of a category
Let $A$ be a dg- or $A_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH_*(A)$ be the Hochschild homology of $A$.
Suppose that $HH_n(A)=0$ for all $n ...
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2
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596
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
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1
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685
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The state of the art on topological rings - the Jacobson topology
I was recently studying the Jacobson density theorem and I found it quite interesting.
Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
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2
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585
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Is the equational theory of groups axiomatized by the associative law?
Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory ...
- 2,023
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1
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448
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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
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5
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Examples of left reversible semigroups
I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See this wikipedia page for the definition of a left ...
- 550
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2
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577
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Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
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