All Questions
1,123 questions
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240
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Using group presentation for its corresponding semigroup?
Somewhere Colin M. Campbell noted:
If $A$ is a semigroup defined as $$A=Sg(\pi)=\langle a_1,\cdots, a_d\mid u_1=v_1,\cdots,u_e=v_e\rangle $$ then the same generators with the same relations can ...
- 233
0
votes
1
answer
403
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When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
0
votes
2
answers
284
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 141
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votes
1
answer
59
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Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
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votes
1
answer
72
views
Reference request: Left $R/k$-modules [closed]
In the paper titled:
On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra
I found the following definition:
Let $k$ be a commutative ring with ...
0
votes
1
answer
128
views
What are the semigroups in which congruence classes can be multplied like sets?
For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and ...
- 1,445
0
votes
2
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273
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Identifying a sequence of polynomials
Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials :
$$X_1$$
$$q^{-1/2}(X_1X_2-1)$$
$$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$
$$q^{-1}(X_1X_2X_3X_4-X_3X_4-...
0
votes
1
answer
352
views
Uniqueness of maximal subfields
Let D be a division ring with center Z. Let R and K be two maximal subfields of D, both purely inseparable of exponent one ( means the p power of each of them in Z).
Why are R and K isomorphic?
Or a ...
- 9
0
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2
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219
views
Are there any central simple algebras admitting a standard basis?
Are there any central simple algebras admitting a standard basis?
By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...
- 113
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1
answer
198
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Example of noncommutative central reduced rings which is not reduced
A ring $R$ is called central reduced if every nilpotent element is central. Ungor et al. math.RA 14 Dec 2013 has given an example of a commutative ring which is central reduced but not reduced. Can we ...
- 585
0
votes
1
answer
400
views
Non-simple and non-unital rings with trivial centres
Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.)
It is not difficult to show that if $R$ is a simple ...
- 1,065
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votes
1
answer
232
views
Kernel elements for the Grothendieck group map of a commutative monoid
This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
- 8,512
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1
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49
views
More vocabulary for periodic elements in monoids
Let $M$ be a monoid, and let $x\in M$. One says that $x$ is periodic if
$$x^{i+j}=x^j$$
for some integers $i\geq 1$ and $j\geq 0$.
An easy division algorithm argument shows that if $m$ is the ...
- 18.7k
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1
answer
655
views
How to show two semigroups are isomorphic?
I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the ...
- 233
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1
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153
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C*-algebra of free monogenic inverse semigroup
Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...
- 493
0
votes
1
answer
381
views
Maximal group image
How does one prove: if $S$ is a finitely generated Clifford semigroup its maximal group image is actually $S_{e_{n}}$?
- 1
0
votes
1
answer
285
views
A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 113
0
votes
1
answer
143
views
Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 73
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1
answer
139
views
Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 1
0
votes
1
answer
200
views
The rank of a semigroup
Let $S$ be a finite noncommutative semigroup(without identity) with a subset $M$ such that $\langle M \rangle =S$. If every element of $M$ is indecomposable in $M$, i.e. for any $a \in M$, there are ...
0
votes
1
answer
296
views
Multiplicative monoid of ring modulo units
Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio.
We define the equivalence relationship $x\...
- 943
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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
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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
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votes
1
answer
83
views
Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials
Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand ...
- 173
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votes
1
answer
305
views
Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
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1
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94
views
What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?
http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...
- 1
0
votes
1
answer
117
views
Local algebras with small maximal left ideals
Is there an infinite-dimensional, non-commutative complex local algebra $A$ (which is not a field) with the (unique) maximal left-ideal finitely generated as a left ideal? Or as a right ideal?
- 43
0
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1
answer
147
views
Small set of acts over a countable monoid?
Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?
- 11
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2
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172
views
small extensions of the free semigroup of rank 1
Let N denote the free semigroup of rank 1. Say that a semigroup T is a small extension
of N if N embeds in T and |T - N| is finite. Is there some kind of classification
of small extensions of N? ...
- 91
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0
answers
61
views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
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0
answers
32
views
Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 923
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0
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92
views
Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
0
votes
0
answers
104
views
Non-degenerate bilinear pairing of finite dimensional algebras
A finite dimensional algebra (over $\mathbb{C}$, say) is said to be Frobenius if it comes equipped with a nondegenerate bilinear pairing
\begin{align*}
\langle -, - \rangle : A \times A \rightarrow \...
- 71
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0
answers
45
views
projections on minimal left ideals of semisimple algebras
Let $KG$ be a semisimple group algebra of a finite group $G$ over $K$. Consider $W=KGe$ as a minimal left ideal of this algebra and $e$ as a primitive idempotent. Here, $W$ is a simple left $KG$-...
- 143
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votes
0
answers
114
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Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
0
votes
0
answers
95
views
Which algebraic structure characterizes the set of non-trivial qudratic residues in a finite field?
I understand this question may be too naive to ask, but I am unable to figure it out.
Suppose, $\mathbb{QR^*}$ denotes the set of all quadratic residues in a finite field except the identity element $...
0
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0
answers
92
views
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
Given that there are $3{,}684{,}030{,}417$ semigroups of order $8$, I guess $n\...
- 379
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0
answers
63
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A construction that sort of merges two semigroups to build a new one
Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
- 10.5k
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0
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122
views
Is there a name for this condition on a monoid?
Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
- 1,951
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0
answers
41
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Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
- 41.9k
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0
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62
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To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
0
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0
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58
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An action on multiplicatively antisymmetric matrix
A matrix $ Q=(q_{ij})$ is called multiplicatively antisymmetric over a field $ F $ if $ q_{ii}=1 $ and $ q_{ij}={q_{ji}}^{-1} $.Let $ \mathcal{Q} $ be the set of all $ n \times n $ multiplicatively ...
- 923
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0
answers
39
views
Countably infinite monoids with minimal right ideals
Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
- 237
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0
answers
293
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Quotient of monoids and monoid algebras
Let $ X $ be a monoid and $ R $ be a (two-sided) congruence relation on $ X $ which is generated by some relations $ u_i \equiv_R v_i $ for any $ i $ in some index set $ J $. Let $ K $ be a field, $ K[...
- 327
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0
answers
105
views
Flag variety as monoid and Schubert calculus
The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation.
When looking ...
- 11
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votes
0
answers
250
views
Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
0
votes
0
answers
43
views
Can Q(R) embed to Q((R ⊗ S )/ P)
Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian.
let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...
- 1
0
votes
0
answers
41
views
Characterizing centralizer of nilpotent self-maps
Let $\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(...
- 109
0
votes
0
answers
96
views
Gelfand–Kirillov dimension of the first Weyl algebra by using the definition
$\DeclareMathOperator\GKdim{GKdim}$Here I am trying to find the Gelfand–Kirillov dimension of the first Weyl algebra just by using the definition of the Gelfand–Kirillov dimension.
Let $A$ be an ...
- 131
0
votes
0
answers
93
views
A semifield of characteristic zero may have a finite number of elements
A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$.
I ...
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