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Tagged with or semigroups-and-monoids semigroups-and-monoids
606 questions
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Name for condition on map of cancellative monoids
Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that
$k(\epsilon)=\epsilon$
for all $a,b\in M$, there exists $v\in N$ such that ...
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103
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Lower periodic subsets of groups and semigroups
Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower $B$-...
- 995
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143
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on reductive monoids which are gorenstein
Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore http://www.cmat.edu.uy/cmat/docentes/alvaro/...
- 3,472
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327
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Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
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152
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Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 481
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2
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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
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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
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283
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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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128
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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
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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 ...
- 621
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232
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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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49
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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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655
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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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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
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381
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Maximal group image
How does one prove: if $S$ is a finitely generated Clifford semigroup its maximal group image is actually $S_{e_{n}}$?
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285
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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
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143
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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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139
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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 $...
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200
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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 ...
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296
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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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63
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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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83
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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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305
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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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94
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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
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147
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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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172
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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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61
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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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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 ...
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95
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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 $...
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92
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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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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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122
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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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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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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 $\mathbb CG$. ...
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39
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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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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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105
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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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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] ...
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41
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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
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92
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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 ...
- 203
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84
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Primage structures: induced domain partitioning by itterated inverse (reference request)
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example, the j-th such preimage list ...
- 23
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213
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make me idempotent
$T_n$ be the full transformation semigroup on $X_n= \{1, 2, \cdots , n\}$.
$D_r =\{\alpha \in T_n: |im(\alpha)|=r\}$.
$E(D_r)$ is the set of all idempotents of semigroup $T_n$.
$support(\alpha)=\{...
- 109
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136
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Amalgamated free-product of semigroups (definition)
I am self-studying some concepts including the title one. I reached the definition of an amalgamated free-product ${S_1}{*_U}S_2$ where $[S_1, S_2; U, w_1,w_2]$ is an amalgam of semigroups. Let $S_1=\...
- 233
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53
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a generalization of group (monoid with order-by-order invertible elements)
Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
- 2,232
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101
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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
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75
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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
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125
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Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual ...
- 10.5k
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0
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136
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Monoid action on an uncountably infinite set
The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition system?)....
- 101
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72
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Decomposition results for locally commutative semigroups
Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
- 798
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179
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semigroup actions of groups on regular rooted trees
If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
- 125