All Questions
1,123 questions
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Conjugate linear maps between $*$-algebra modules
Let $A$ be a $*$-algebra, $E,$ and $F$ two $A$-modules, and a map $f:E \to F$ such that
$$
f(ae) = a^*f(e), ~~~~~~~ a \in A.
$$
This seems to me to be the natural generalisation of a conjugate linear ...
- 1,479
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1
answer
260
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The intersection of Block Groups and R-trivial (finite) monoids
Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the ...
- 424
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1
answer
636
views
Local Rings problem
$\newcommand{\End}{\operatorname{End}}$
let $R$ be a local ring, $\varphi\in \End(R_{R}^{2})$,
$\overline{\varphi}\in \End(\overline{R}_{\overline{R}}^{2})$,
$\overline{R} =R/J(R)$ , $J(R)$= ...
- 11
1
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1
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87
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Semigroup algebras with one dimensional center
Let $S$ be a finite semigroup and $K$ a field of characteristic 0 (we can assume the complex numbers for simplicity).
Question: Is there a characterization when the center of the semigroup algebra $...
- 26.5k
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1
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129
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Cycles in almost breakable semigroups
Last October, I learned from Benjamin Steinberg's answer to another question of mine that a semigroup $S$ is called breakable if $xy \in \{x, y\}$ for all $x, y \in S$. Let's now say that $S$ is an ...
- 10.5k
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1
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98
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If, in a unit-regular ring, the right annihilator of $a$ equals the right annihilator of $b$, then $aR = bR$?
Recall that a (unital) ring $R$ is von Neumann regular (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and unit-regular if such an element $y$ can be taken to be a unit.
...
- 10.5k
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1
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85
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Derivable relations in a monoid
Let $ X $ be a monoid which is generated by the elements $ x_1, x_2, \hat x_1, \hat x_2 $ and the relations $ \hat x_i x_i = 1 $ and $ x_i \hat x_j = \hat x_j x_i $ for any distinct $ i, j = 1, 2 $.
...
- 327
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1
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147
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Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...
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1
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326
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Closed submonoid of $(\mathbb{C}^*)^n$
The answer of this question might be known but I was not able to find any answer. Let $n\geq 1$ and $S$ be a closed submonoid of $(\mathbb{C}^*)^n$, that is, a closed and stable by product subset of $(...
- 143
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1
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70
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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
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1
answer
110
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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
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1
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158
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Semigroups admitting commutative group actions
Let $(S,*)$ be a semigroup admitting a distinguished element $0$ such that $z*s = s*z = z$, for all $s \in S$. Moreover, let $(\mathbb{G},\cdot)$ be a commutative group. Consider an action
$$
\mathbb{...
- 219
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1
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317
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Monoids (or semigroups) with a "finite decomposition" property
In my research I have come across the following condition on a monoid.
Every element $x$ satisfies the following property: there exists a natural number $n$ such that for any $m \geq n$ and any ...
- 408
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1
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52
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When is $rad(L)[x_1,\ldots]$ radical in $Ker(\varphi_\ast)$?
Suppose we have a local ring $L$ (not necessarily commutative) such that $L/rad(L)$ is a division algebra (here $rad(L)$ is the Jacobson radical of $L$). We clearly have the canonical surjection $\...
- 98
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1
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196
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Lax monoids where only the unit triangle is lax
I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...
- 327
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1
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166
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Integral elements of quaternion algebras with predescribed properties
In the course of doing some calculations I have found myself wanting to answer the following question:
Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let $\...
- 562
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1
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72
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Transformation terminology question
Given a transformation $t$ from the transformation semigroup $T_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a ...
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2
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369
views
Gerstenhaber bracket out of $L_\infty$ algebras
Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that $\...
- 41
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1
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216
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Counting modular squares in an interval
For an integer $m$, let $S^m_{x_0,x_1} = \{ t | x_0 ≤ t ≤ x_1 $ and $t$ is a square modulo $m \}$. Let $S^m_x$ = $S^m_{0,x}$.
Determining whether the sets $S^m_x$ are empty is easy (1 is always a ...
- 123
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1
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130
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Projective dimension over hypersurface
Let $R$ be (not necessarily commutative) ring and $S$ a simple right $R$-module. Let $f\in Ann(S)$ be normalizng and a non-zero divisor. Is it always true that
$$
pdim_{R}(S)=pdim_{R/(f)}(S)+1?
$$
- 63
1
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1
answer
154
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undecidability in the dynamics of functions $f: \Sigma^* \rightarrow \Sigma^*$
Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^* to \Sigma^* where \Sigma is a finite set? In particular, I'm ...
- 549
1
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1
answer
274
views
Q-Divisor and Determinant Map on a Maximal Order
Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...
- 1,391
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0
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105
views
What is a quantum condensed space?
Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
- 1,037
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0
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30
views
Star-algebra isomorphism
I have asked this question: When an algebra isomorphism preserves positive involution, but now I want to modify it.
Let $A$ and $B$ be $K$-algebras where $K$ is a field with a unique ordering.
We say ...
- 143
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0
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90
views
Multiplicative bases, path algebras, and Ext algebras
I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional.
It ...
- 71
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0
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69
views
On a lemma of projective dimension
Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...
- 157
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84
views
Simple-direct-injective modules
A right $R$-module $M$ is called a simple-direct-injective module if it satisfies any of the following equivalent conditions:
For any simple submodules $A,B$ of $M$ with $A \cong B \subseteq^{\oplus} ...
- 324
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0
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86
views
Inner product on Standard form of von Neumann algebra
Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...
- 131
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0
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35
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Number of right divisors of a central skew polynomial
Let $\mathbb{F}$ be a finite field of $p$ elements, $\sigma \in \operatorname{Aut}(F)$ of order $m$, $\mathbb{F}^\sigma$ be the fixed field of $\sigma$, and $\mathbb{F}[x,\sigma]$ be a skew polynomial ...
- 323
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0
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69
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Structure of tame concealed algebra of Euclidean type
I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
- 839
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108
views
Does the center of any finitely generated associative algebra over a field have finite type?
Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
- 331
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48
views
Neighborhoods of idempotents in topological inverse semigroups
In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
- 1,093
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0
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95
views
Eventual stabilization for repeatedly adding multiplayer games
This question is an outgrowth of a couple previous questions of mine. In order: 1,2,3. This should be fully self-contained, but those questions may help motivate this one.
To keep things readable, I'...
- 20.5k
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0
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91
views
A term for a submonoid of a free abelian monoid?
Are there multiple ways of characterising which monoids are submonoids of free abelian monoids?
What free abelian monoids are:
A free abelian monoid $\mathbb N^d$ with $d$ generators (where $d$ is an ...
- 4,943
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0
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138
views
Category whose morphisms are commutative monoids but not enriched
In a recent investigation, I constructed a category $\mathcal{C}$ with the following property. For objects $X,Y \in \mathcal{C}$, the morphism set $\text{Mor}(X,Y)$ is a commutative monoid with ...
- 161
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0
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70
views
Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
- 883
1
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0
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274
views
Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,306
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87
views
Abelianization of the group of invertible elements in a finite local ring
Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{\mathrm{ab}}$?
(We can factor $R$ be ...
- 5,039
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0
answers
53
views
The "hyperbolicity preserving" probabilities
A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of
$$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
- 52.3k
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156
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Pseudo-coherent complexes over sheaves of non-commutative rings
I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.
Assume that $\mathcal{R}_X$ is a ...
- 609
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0
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43
views
Interleaving in Viennot's Heaps models?
I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
- 11
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0
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85
views
Exponential of a sum in a non-commutative graded algebra
Let $a,b$ be two elements of a graded algebra $A$ such that $\deg(a)=1$, $\deg(b)=0$ and $[a,b]\neq 0$.
I would like to know whether there exits an explicit expression for the degree 1 component
$$\...
- 789
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88
views
Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
- 1,547
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0
answers
56
views
Effect on finite transformation semigroup under a particular modification of the generators
The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
- 695
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0
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66
views
First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?
In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
- 1,093
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0
answers
170
views
What is this algebraic object (special case of a semigroup)?
Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
- 696
1
vote
0
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39
views
Rings where every indecomposable principal right ideal is extensive
Let $R$ be a (commutative or non-commutative, associative) unital ring. Following Nicholson & Yousif [1, p. 21], we say that a right ideal $\mathfrak i$ of $R$ is extensive if every $R$-linear ...
- 10.5k
1
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0
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124
views
On the rings $R$ with the property that $eR \cong fR$ for all primitive idempotents $e, f \in R$
Let $R$ be a (commutative or non-commutative) ring with identity. As usual, an idempotent $e \in R$ is primitive if $eR$ (the principal right ideal generated by $e$) is indecomposable as a right ...
- 10.5k
1
vote
0
answers
139
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k
1
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0
answers
121
views
Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as
$$
\mathcal{U} * \mathcal{V} = \left\{ A \...
- 59