All Questions
1,123 questions
3
votes
1
answer
246
views
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
votes
1
answer
203
views
Centralizer of a single element in the monoid of self-maps of a set
This is a follow-up to this question: For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
Let $X$ be a set, and $X^...
- 63.9k
3
votes
1
answer
408
views
Finite Homological Dimension of R/P for all P for module finite non-commutative rings
I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's ...
- 3,758
3
votes
1
answer
394
views
Residual finiteness of groups versus residual finiteness of semigroups
A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't ...
- 155
3
votes
1
answer
311
views
Functors that preserve monoids
In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...
- 499
3
votes
1
answer
244
views
Category of continuous self maps
Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?
How can we tell whether a category is the category of continuous ...
- 2,772
3
votes
1
answer
128
views
Size of a minimum generating set for full transformation monoids
Given any finite set $X$ the set $\mathcal{T}(X)=X^X$ of all functions from $X$ to $X$ clearly forms a monoid under composition. Now if we call any family of functions $\mathcal{F}\subseteq \mathcal{T}...
- 2,506
3
votes
2
answers
215
views
What do we call functions satisfying $[a[b]c] = [abc]$?
Let $M$ denote a monoid and suppose we're given a function $[-] : M \rightarrow M$ satisfying $[a[b]c] = [abc].$ Then:
Proposition 0. $[-]$ is idempotent.
Proof. Take $a=c=1$).
Proposition 1. ...
- 3,793
3
votes
3
answers
345
views
Examples of cancellative normal semigroups
I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in S\...
- 31
3
votes
1
answer
243
views
Embedding Semigroups in Rings
Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is ...
- 801
3
votes
1
answer
297
views
Is the universal inverse semigroup of a commutative semigroup an embedding?
The question of existence of a universal inverse semigroup of an arbitrary semigroup has been answered before (this is a construction similar to the Grothendieck group). Let's refer to the universal ...
- 2,464
3
votes
1
answer
706
views
Center of universal enveloping algebra of nilpotent lie algebra
Let g be a finite dimensional nilpotent lie algebra over a field k of characteristic zero. Let U(g) be the universal enveloping algebra and Z(g) be its center. Denote by Z_1(g) the augmentation ideal ...
- 3,758
3
votes
1
answer
350
views
Special subalgebras of central simple algebras
In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is ...
- 279
3
votes
1
answer
229
views
A pexiderization of the sine addition law on semigroups
Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?
3
votes
2
answers
324
views
An integral transform and the Stone-Weierstrass theorem
For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if
$$
\int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
- 303
3
votes
1
answer
552
views
Lax monoidal functor
Let me denote $Cat$ the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let $F: (Cat, \times)\rightarrow (Set,\times)$ a symmetric monoidal ...
- 223
3
votes
1
answer
157
views
Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?
Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
- 6,962
3
votes
1
answer
404
views
How many monoids with $n$ arrows exist?
How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
- 41
3
votes
1
answer
199
views
group completion theorem by using homology fibrations
In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
- 2,223
3
votes
1
answer
285
views
Cancellative semigroup on a distributive lattice
Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?
- 1,145
3
votes
1
answer
220
views
Intersection of Maximal Left Ideals with Finite Dimensional Quotient
Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I \...
- 3,031
3
votes
2
answers
477
views
noncommutative polynomials equality
Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.
Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...
- 1,528
3
votes
1
answer
226
views
Presentation of the monoid of self-maps of a finite set
Is there a common presentation of the semigroup of functions from a given (finite) set to itself?
- 53
3
votes
1
answer
367
views
submonoid of a matrix monoid with a common eigenvector
Hello,
I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ ...
- 69
3
votes
1
answer
137
views
An f.g.u. duo monoid is unit-duo: True or false?
Let $H$ be a monoid (written multiplicatively) with the property that $H = H^\times A H^\times$ for some finite $A \subseteq H$ (shortly, an f.g.u. monoid), where $H^\times$ is the group of units of $...
- 10.5k
3
votes
1
answer
173
views
Well-foundedness of divisibility vs well-foundedness of right- and left-divisibility
Say that a preorder (i.e., a reflexive and transitive binary relation) $\preceq$ on a set $X$ is
artinian if there is no sequence $(x_n)_{n \ge 1}$ of elements of $X$ with $x_{n+1} \prec x_n$ for ...
- 10.5k
3
votes
1
answer
197
views
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=\...
- 131
3
votes
1
answer
252
views
Classification of associative polynomial functions
What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers ...
- 6,962
3
votes
2
answers
249
views
Question about actions of full transformation monoids
[Reposted from math.stackexchange]
Consider a monoid $M$ acting on a set $X$, where $M$ is the full transformation monoid on some set $A$ (i.e., the set of all functions from $A$ to itself, with ...
- 167
3
votes
1
answer
188
views
Finite congruence-free semigroup without zero [closed]
I am reading the book " Fundamentals of semigroup theory" by John M. Howie $\textbf{(Section 3.7)}$.
I want to prove $\textbf{Theorem 3.7.2}$
If $S$ is a finite congruence free semigroup ...
- 153
3
votes
2
answers
101
views
A non-reduced, commutative BF-monoid s.t. $au = u$ for all $a \in \mathcal A(H)$ and $u \in H^\times$
Let $H$ be a monoid, and denote by $H^\times$ and $\mathcal A(H)$, respectively, the set of units (or invertible elements) and the set of atoms (or irreducible elements) of $H$ (an element $a \in H$ ...
- 10.5k
3
votes
2
answers
476
views
Rings all of whose torsion modules are cyclic
Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring:
Let $k=\mathbb{C}((t))$ and let $R=k[\...
- 2,751
3
votes
1
answer
181
views
Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?
A Lie algebra $\mathfrak{g}$ generates its universal enveloping algebra $\mathrm{U}\mathfrak{g}$, which has the structure of a Hopf algebra. Modules of $\mathrm{U}\mathfrak{g}$ are exactly the of ...
- 183
3
votes
1
answer
310
views
Balanced dualizing complex vs rigid dualizing complex?
In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing ...
- 1,663
3
votes
1
answer
529
views
Study of free monoids of the recursive S. Eilenberg.
Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...
- 4,165
3
votes
1
answer
171
views
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational
Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
- 10.5k
3
votes
1
answer
376
views
Graded global dimension of a graded algebra
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...
- 889
3
votes
2
answers
172
views
On the Hilbert function of a numerical semigroup
Recall that a numerical semigroup $S$ is a submonoid of the non-negative integers $\mathbb Z_{\geq 0}$ whose relative complement $\mathbb Z_{\geq 0} \setminus S$ is finite. Observe that the collection ...
- 183
3
votes
1
answer
182
views
Can one turn finite-dimensional vector subspaces into a cancellative semigroup?
Let $V$ be a vector space over some field and let ${\rm Fin}\,V$ be the family of all finite-dimensional subspaces of $V$. Is it possible to turn ${\rm Fin}\,V$ into an commutative cancellative ...
- 31
3
votes
1
answer
149
views
Question on monoid algebras
Let $G$ be a finite monoid.
Question 1: In case the monoid algebra $A=kG$ is weakly symmetric (meaning soc(P)=top(P) for each indecomposable projective modules), is $kG$ even symmetric (meaning $A \...
- 26.5k
3
votes
1
answer
125
views
Quasi-isometries and E-unitary inverse semigroups
Let $S = \langle K\rangle$ be a finitely generated inverse semigroup, where $K \subset S$ is a fixed, finite and symmetric set of generators.
Preliminaries: Recall that we say that $s, t \in S$ are $\...
- 733
3
votes
1
answer
233
views
Is there any characterization and/or classification of subsemigroups of finite monogenic semigroups?
A semigroup $S$ is called monogenic if $S$ is generated by some element $a$ (which is unique if $S$ is not a group) in the sense that $S=\{a^n:n\in\mathbb N\}$.
Observe that each mongenic group is ...
- 41.9k
3
votes
1
answer
313
views
Are local rings of monoid algebras geometrically unibranch?
Let $\mathrm{M}$ be a finitely generated submonoid of $\mathbb{Z}^{\oplus d}$ for some $d$, let $A := k[\mathrm{M}]$ be the associated monoid algebra over a field $k$, let $\mathfrak{m} \subset A$ be ...
- 2,017
3
votes
2
answers
165
views
Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality
Let $H$ be a multiplicatively written, commutative monoid with identity $1_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
- 10.5k
3
votes
1
answer
122
views
A BF-monoid $H$ s.t. $H^\times$ is not divisor-closed
Let $H$ be a (multiplicative) monoid, and denote by $H^\times$ the set of units of $H$ and by $\mathcal A(H)$ the set of atoms of $H$ (let me recall that an element $a \in H$ is an atom if (i) $a \...
- 10.5k
3
votes
1
answer
101
views
Isomorphism concerning $Soc(M_n(R))$
It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
- 355
3
votes
1
answer
190
views
What is known about semigroups that are generated by (cyclic) subgroups?
A semigroup $(A,\cdot)$ that is idempotent (i.e. $a^2=a$ for every element $a\in A$) is naturally generated by its subgroups (every element on itself constitutes a trivial group). I would like to know ...
3
votes
1
answer
212
views
Liftability of a submodule from an associated graded module
Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$.
Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, ...
- 27.9k
3
votes
1
answer
875
views
The Jacobson radical of an infinite dimensional algebra
Does any one know the Jacobson radical of the path algebra of the following quiver?
$$\bullet \leftrightarrows \bullet$$
How many simplerepresentations of it are there?
Is there any software that ...
- 207
3
votes
1
answer
392
views
Can cones (toric monoids) be built as colimits of their faces?
Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...
- 24.1k