All Questions
1,123 questions
2
votes
1
answer
116
views
On the maximum elements of a numerical semigroup that have order between $n$ and $2n$
Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
- 183
2
votes
1
answer
287
views
On the definition of the Cherednik algebra of a variety with a finite group action
Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...
- 541
2
votes
1
answer
246
views
The combinatorics of $(f \partial)^n$ in the noncommutative setting?
This is a noncommutative version of these three previous questions:
differential operator power coefficients
Сlosed formula for $(g\partial)^n$
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
...
- 7,127
2
votes
1
answer
67
views
$E$-separated semigroups
Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is ...
- 41.9k
2
votes
1
answer
166
views
Submonoid of free monoid with certain properties
Let $N$ be a submonoid of a free monoid $M$ such that
$m_1nm_2\in N \Rightarrow m_1,m_2\in N$ for any $m_1,m_2\in M$ and $n\in N\setminus\{1\}$. $\quad\quad\quad\quad$ (C)
Do such submonoids ...
- 491
2
votes
1
answer
287
views
Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable
Crossposted on StackExchange on July 28 (no answer so far).
Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
- 10.5k
2
votes
1
answer
192
views
Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR
Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ...
- 10.5k
2
votes
1
answer
148
views
Terminology for a ring where every right cancellable element is cancellable
Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
- 10.5k
2
votes
1
answer
404
views
Reference request: a cousin to the log semiring
Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
- 15.4k
2
votes
1
answer
153
views
Define a homomorphism of a set of graphs to its power set
Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is,
$G_1\cup G_2$
$=\langle V(G_1)\cup V(G_2), (E(G_1)\...
- 203
2
votes
1
answer
195
views
A question about semigroup union
The semigroup of all order-preserving and decreasing transformations in full transformations semigroup $T_n$ is denoted $C_n$.
I consider the idempotent set $A=\{\begin{bmatrix}2\\1 \end{bmatrix},\...
- 355
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
user111492
2
votes
1
answer
122
views
If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?
I will first state my question, and then give all the relevant definitions.
Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
- 10.5k
2
votes
1
answer
368
views
Finding index/period of a semigroup element
The index and period of a finite monogenic semigroup $\langle x\rangle$ are the smallest numbers $i$ and $p$, respectively, satisfying $x^{i+p}=x^p$. The question is:
Is there an algorithm to find ...
- 1,651
2
votes
1
answer
211
views
Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit
Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied?
$$\text{(P) If }\,xy = x\...
- 10.5k
2
votes
1
answer
515
views
dg-resolution of the polynomial algebra
I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative).
The resolutions ...
- 165
2
votes
1
answer
294
views
Finitely generated ordered monoids and noetherian subsets
(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
- 6,700
2
votes
1
answer
223
views
pseudovarieties and profinite group : do * and g() commute?
Let $V$ and $W$ be pseudovarieties of finite monoids. We denote with $gV$ the pseudovariety of categories generated by $V$, and by $V*W$ the semidirect product of pseudovarieties $V$ and $W$.
Does ...
- 421
2
votes
1
answer
578
views
Interpretation of Hochschild Homology groups
In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc....
In ...
- 5,405
2
votes
1
answer
132
views
Maximal sub-inverse semigroups of $M_n(\mathbb{C})$ and $M_n(F_p)$
An inverse semigroup $S$ is a semigroup in which every element $x \in S$ has a unique inverse $y \in S$ such that $x = xyx$ and $y = yxy$. Are there some references characterizing the maximal sub-...
- 6,201
2
votes
1
answer
290
views
Idempotents in Green J classes
I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...
- 233
2
votes
1
answer
796
views
Commutative, idempotent partially ordered monoids
A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for commutativity). ...
- 2,442
2
votes
1
answer
164
views
Algorithmically finite-dimensional (noncommutative) algebras.
Can anyone help to find some information about these structures?
2
votes
1
answer
1k
views
monoid ring and some structure within it - how is it called?
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
- 1,626
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
2
votes
0
answers
95
views
Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 1,013
2
votes
0
answers
66
views
Noncommutative transcendence degree of representation algebras
Let $G$ be a reductive group, for example $\text{GL}_n(\mathbb{C})$. Let $V$ denote its defining representation, and let $R$ denote the tensor algebra on the irreducible representations of $G$. It may ...
- 2,907
2
votes
0
answers
69
views
Is anything known about the center of the Fomin-Kirillov algebra?
Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...
- 1,066
2
votes
0
answers
80
views
An alternative definition for finitely generated (and principal) ideals in a semigroup
Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
- 10.5k
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
2
votes
0
answers
144
views
Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$
Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.)
(Please choose any irrep ...
- 21
2
votes
0
answers
71
views
Classification of polynomials leading to finite dimensional admissible algebras
Let $K \langle x , y \rangle $ ($K$ a field, we can assume it has only two elements if it helps) be the non-commutative polynomial ring in 2 variables.
Question 1: For which non-commutative ...
- 26.5k
2
votes
0
answers
176
views
On the origin of power semigroups
Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty ...
- 10.5k
2
votes
0
answers
111
views
Correct notion of "connected" for dga of bundle-valued forms
Consider a vector bundle $E$ over a manifold $M$ with flat connection, $\nabla$. From this data I can form the associative/unital differential graded algebra $\mathcal{A} = \left(\Omega^{\bullet}(M, ...
- 1,611
2
votes
0
answers
68
views
Semigroups related to iterated orthogonal complement
Let $R\subset V\times V$ be a relation on a set $V$. For a subset $S\subset V$, define its orthogonal complement with respect to $R$
as
$$S^l:=\{ x: \forall y\in S\ \ (x,y)\in R\},\ \ S^r:=\{y: \...
- 513
2
votes
0
answers
64
views
A particular generalization of free partially commutative monoids
A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
- 21
2
votes
0
answers
73
views
What should I call a log scheme with free reduced monoids?
This is a terminology question about a class of log varieties.
Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
- 7,952
2
votes
0
answers
161
views
Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
user491484
2
votes
0
answers
181
views
So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
2
votes
0
answers
172
views
Simple modules of quantum planes
Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...
- 889
2
votes
0
answers
88
views
The generators of twisted homogeneous coordinate rings
Let $X$ be a projective scheme over an algebraically closed field $k$ of characteristic $0$.
Let $\sigma$ is an automorphism of $X$ and $\mathcal{L}$ be an invertible sheaf on $X$.
Let $B := B(X, \...
- 889
2
votes
0
answers
101
views
What is the relationship between ramification in central simple algebras and in fields?
Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...
- 121
2
votes
0
answers
67
views
Type of numerical semigroups is not bounded when embedding dimension is $\geq 4$
I am currently studying numerical semigroups. I know that there is no upper bound for the type of a numerical semigroup with embedding dimension greater or equal than $4$. There is a famous example by ...
- 121
2
votes
0
answers
174
views
Moduli spaces of stable sheaves on noncommutative projective schemes
In noncommutative algebraic geometry in the sense of Artin and Zhang, can we construct moduli spaces of stable sheaves on noncommutative projective schemes as (commutative)schemes ?
I would appreciate ...
- 889
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 31
2
votes
0
answers
73
views
Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
2
votes
0
answers
145
views
Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 2,605
2
votes
0
answers
119
views
The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
2
votes
0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
votes
0
answers
98
views
Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?
Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
- 10.5k