All Questions
1,123 questions
2
votes
0
answers
51
views
Existence of a canonical embedding from a quotient of $\mathscr{F}^\ast(\mathcal A(H))$ to another
Let $H$ be a multiplicative, commutative monoid, and denote by $H^\times$ the group of units of $H$, by $\mathcal A(H)$ the set of atoms (or irreducible elements) of $H$, by $\mathscr{F}^\ast(\mathcal ...
- 10.5k
2
votes
1
answer
169
views
A completely simple semigroup with cancelation is a group (simple proof)
Is there a simple proof of the following fact:
Fact. Let $S$ be a completely simple semigroup with cancellations, i.e. each of the equalities $xa=xb$, $ax=bx$ implies $a=b$. Prove that $S$ is a group....
- 153
0
votes
1
answer
183
views
Right localization of $R[x,x^{-1}]$ at monic $f\in R[x]$
Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\...
- 348
2
votes
1
answer
105
views
Localising a right Noetherian ring at a set of regular elements
Let $R$ be a right Noetherian ring, and $S$ a multiplicative set consisting of regular elements where $1\in S$ and $0\not\in S$. Does the right ring of fractions $RS^{-1}$ exist?
This is what I know ...
- 348
3
votes
0
answers
96
views
Given a primitive finite set $A\subseteq\bf N$ with $0\in A$, find two more primitive sets $B,C\subseteq\bf N$ with $B\ne C$ and $A+B=A+C$
Let $\mathcal P_{{\rm fin},0}(\mathbf N)$ be the monoid of all finite subsets of $\mathbf N$ containing $0$ with the operation of set addition
$$
(X, Y) \mapsto X + Y := \{x+y: x \in X \text{ and }y \...
- 10.5k
3
votes
0
answers
92
views
On the eventually regular monoids and generally regular monoids
Planning the problem:
First we give some definitions.
An element $s \in S$ is called eventually regular if for every $s \in S$ there exist a natural number $n$ in $\mathbb{N}$ and $x
\in S$ such ...
- 601
0
votes
0
answers
101
views
Spherical Rings
My question is concerned with filtered rings. It is a classical result that if $R$ is a finitely generated commutative ring graded by a semigroup $S$ then $S$ is also finitely generated.
The reverse ...
- 501
2
votes
0
answers
139
views
A certain non-clean ring
I am searching for a non-commutative ring $R$ with identity such that $R$ is not a clean ring and $R/Soc(R_R)$ is a Boolean ring. By a clean ring I mean a ring each of whose elements is a sum of a ...
- 355
3
votes
1
answer
101
views
Isomorphism concerning $Soc(M_n(R))$
It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
- 355
0
votes
0
answers
75
views
Homomorphic image of $B_{\lambda}^o(S)$ is the Brandt $\lambda^o$-extension of some monoid with zero
Let $S$ be a monoid with zero and $I_{\lambda}$ be an indexed set, then $B_{\lambda}(S) = \{ (\alpha, s , \beta ) : \alpha , \beta \in I_{\lambda}, s\in S \} \cup \{0\}$ is a semigroup and $J = \{ (\...
- 143
6
votes
1
answer
259
views
are endomorphisms "small" compared to the full transformations?
$\DeclareMathOperator\End{End}$Let $T_n$ be the full transformation semigroup/monoid of $[n]=\{1,\dots,n\}$. Let $\End(T_n)$ be the set of [endomorphisms][1] of $T_n$. Then, $\# T_n=n^n$ and
$$\# \End(...
- 43.2k
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
15
votes
7
answers
973
views
Statements about groups proved using semigroups
Question. Has a statement about groups ever been proved using the theory of semigroups?
By "a proof using the theory of semigroups" I do not mean that some steps in the proof are in fact statements ...
14
votes
1
answer
545
views
Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4?
Let $R$ be a commutative ring. Let $A$ be an $R$-algebra (i.e., an $R$-module
equipped with an $R$-bilinear multiplication map that turns $A$ into a unital
ring). We do not require $A$ to be ...
- 33.8k
4
votes
1
answer
91
views
Endomorphism of Brandt Semigroup $B_n(G)$, where $G$ is a finite group
I want to show that $End_0 (B_n(G)) = \cup\phi_{\sigma,g} \cup C_{I(B_n(G))}$, where $\phi_{\sigma,g} : B_n(G) \rightarrow B_n(G) $ is an endomorphism is defined by $(i,a,j)\phi_{\sigma,g} = (i\sigma ...
- 143
15
votes
4
answers
3k
views
Why should the tensor product of $\mathcal{D}_X$-modules over $\mathcal{O}_X$ be a $\mathcal{D}_X$-module?
Let $R$ be a regular algebra over a field $k$ of char 0. Let $D$ be its corresponding algebra of differential operators.
As in the general setting of non-commutative algebra we can tensor right $D$-...
- 7,789
2
votes
0
answers
135
views
Is a ring with stable range 2 2-Hermite?
Let $R$ be a (possibly non-commutative) ring. The left stable range of $R$ (denoted $sr_l(R)$) is the smallest $n$ such that every left unimodular row of length $>n$ is reducible. A similar ...
- 61
1
vote
1
answer
52
views
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
8
votes
1
answer
924
views
What's the cokernel of a monoid homomorphism?
Let $f:A\to B$ be a monoid homomorphism. Where can I find an explicit description of the its cokernel? Are there any books on this topic?
By the cokernel of $f$, I mean the universal arrow which ...
- 10.5k
14
votes
1
answer
1k
views
Category without identities?
Just as a monoid is a category with a single object, a semigroup may be seen as a non-unital category, still with associative composition. Then an $S$-set for $S$ a semi-group can be seen as a functor ...
- 10.5k
2
votes
1
answer
293
views
Notation and reference for polynomials with coefficients not commuting with the indeterminates
Let $R$ be a noncommutative ring (with unit). Then a "fully noncommutative" (for a lack of better wording) monomial over $R$ in the single noncommutative indeterminate $X$ of degree $d$ is given by a ...
- 7,127
-1
votes
2
answers
1k
views
Binomial expansion for noncommutative operator
Is it possible to find a closed formula for $(A^\dagger -kA)^n$ with $[A,A^\dagger]=1$ ?
I am looking for the normal ordinate form: $\sum (A)^{n-j}(A^\dagger)^j$— possibly something to do with the ...
2
votes
1
answer
167
views
Why does the monoid of central morphisms act transitively?
I'm reading and struggling with bits and pieces of the book Mal'cev, Protomodular, Homological, and Semi-Abelian categories by Borceux and Bourn. At the moment I'm having trouble with:
Theorem 1.3.22 ...
- 10.5k
5
votes
2
answers
402
views
Maximal commuting subsets of $\text{End}(X)$
Let $X$ be a set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. We say that $f, g\in \text{End}(X)$ commute if $g\circ f = f\circ g$, and $S\subseteq \text{End}(X)$ is a commuting ...
- 48.9k
2
votes
0
answers
106
views
Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
5
votes
2
answers
236
views
Descent of flatness from algebras to monoids
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
- 6,700
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
0
votes
0
answers
87
views
When does an automorphism extend to a localisation (noncommutative rings)
Let $R$ be a (not necessarily commutative) ring. Let $\tau$ be an automorphism of $R$. Consider the localisation of $R$ at a set of multiplicative elements which satisfy the ore condition, say $X$. ...
- 201
4
votes
0
answers
367
views
Is a central simple algebra necessarily cyclic if it splits after a cyclic Galois extension?
Let $A$ be a central simple algebra of degree $n$ over $k$, $\dim_kA=n^2$, let $K/k$ be a cyclic galois extension of degree $n$. Suppose $A\times_kK\cong M_n(K)$, does this imply that $A$ is a cyclic ...
user39380
0
votes
1
answer
63
views
Monoid morphisms satisfying a decomposition condition
Let $A$ and $B$ be monoids, let $f\colon A\to B$ be a morphism of monoids. The following pair of conditions emerged naturally in my research:
For all $a\in A$ and $b_1,b_2\in B$ such that $f(a)=b_1....
- 327
1
vote
0
answers
639
views
What is the real name of this relation and operation on a particular set of maps between cancellative monoids?
Let $A,B$ be cancellative monoids and define a transducer as a map $f\colon A \rightarrow B$ such that $f(1)=1$ and for all $a_1 ,a_2 \in A$, there exists a $b \in B$ such that $f(a_1 a_2)=f(a_1) b$. ...
2
votes
0
answers
227
views
What is the motivation behind the definition for a smooth differential graded category?
Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\...
- 1,716
2
votes
1
answer
263
views
Nonnegatively graded algebra $A$ finitely generated as $k$-algebra iff $A_0$ finitely generated, $A_{>0}$ finitely generated as $A$-module?
This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated ...
- 349
3
votes
1
answer
173
views
$M$ is finitely generated as $A$-module iff $M/A_{>0}M$ is finitely generated as $A$ module?
Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$.
(Here, given a nonnegatively graded algebra $A$, we've ...
- 349
7
votes
2
answers
309
views
Homological questions on monoid algebras
Given a finite monoid G and its group algebra A over a field $K$.
I have never really studied such algebras, but I have the following questions (which are probably basic questions on any large class ...
- 26.5k
17
votes
0
answers
536
views
Question about combinatorics on words
Let $\{a_1,a_2,...,a_n\}$ be an alphabet and let $\{u_1,...,u_n\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$...
user6976
7
votes
1
answer
350
views
Pushouts of injective monoid homomorphisms
Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and ...
- 5,482
5
votes
1
answer
342
views
Can the trivial module be stably free for a monoid ring?
Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ ...
- 38.6k
6
votes
0
answers
215
views
Matrix semigroups in which a weighted average of eigenvalues is multiplicative
A problem in fractal geometry requires me to consider matrix semigroups with the following curious property. For a $d \times d$ real matrix $A$ let $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \...
- 6,206
4
votes
1
answer
855
views
Left- and right-sided principal ideals of quaternions have same index?
One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
- 5,827
8
votes
2
answers
483
views
Posets obtained from a semigroup by the definition $x \leq y \iff x \cdot y = x$
A po-groupoid is a groupoid $\langle A,\cdot\rangle $ such that the relation
defined by
$$
x \leq y \text{ if and only if } x \cdot y = x
$$
is a partial order on $A$, the order related to $\langle ...
- 1,124
0
votes
1
answer
63
views
What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?
Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$.
What are ...
- 103
5
votes
2
answers
364
views
Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...
- 505
2
votes
1
answer
569
views
Why $k[x,y]$ is not a formally smooth algebra?
We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define
$$
D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar{...
- 9,019
3
votes
1
answer
123
views
Is $T(V) \rtimes T(V^* \otimes V)$ a bialgebra?
Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$.
The algebras $T(V)$ and $T(V^* \otimes V)$ are bialgebras. I am trying to find some bialgebra ...
- 6,211
11
votes
1
answer
740
views
Determinants of octonionic hermitian matrices
For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying
$a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows.
...
- 21.8k
10
votes
1
answer
243
views
Can a semigroup with zero be globally isomorphic to a semigroup without zero?
This is not a great question for sure and it may even be trivial for all I know, but a couple of years ago, when I still thought I'd be a mathematician, I spent quite a lot of time thinking about it ...
- 1,445
6
votes
1
answer
325
views
reduced norm from degree 3 division algebra
Let $D$ be a degree $3$ division algebra over a field $k$ of char not 2 and 3.
Any such division algebra is cyclic. I am interested in knowing the cases when the reduced norm map $Nrd : D^* \...
- 661
5
votes
2
answers
340
views
Gelfand-Kirillov dimension of generalized Weyl algebras
I believe that the Gelfand-Kirillov (GK) dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$.
Does anyone have a reference for this?
I can find partial results, and I ...
- 201
6
votes
1
answer
302
views
Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?
Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$.
By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to Z(...
- 255