All Questions
Tagged with or semigroups-and-monoids semigroups-and-monoids
606 questions
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
0
votes
1
answer
153
views
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
3
votes
0
answers
79
views
Semigroups containing an ideal with a local identity
I'm looking for some classes of semigroups containing a (proper) ideal with a local identity (i.e., ideal submonoid). Can somebody give some examples or/and theorems for the followings cases:
(a) ...
- 995
2
votes
1
answer
130
views
Factorisation of positive definite functions
Let $(S,\circ)$ be a semigroup with identical involution. Of course, we know that products of positive definite functions on $S$ are again positive definite. I'm interested in the other direction, ...
- 289
3
votes
0
answers
137
views
Can monoids of "continuous words" be realized as initial monoid objects?
Whenever $X$ is a set, write $X^*$ for the monoid freely generated by $X$. The elements of $X$ are, of course, words in the letters $X$. When $X$ is finite, there also seems to be a great many ...
- 3,793
2
votes
0
answers
63
views
QF-3 monoid algebras
A finite dimensional algebra $A$ is called QF-3 in case the injective envelope of the regular module is projective. For example all Frobenius algebras are QF-3.
Given a monoid algebra $kG$ of a finite ...
- 26.5k
2
votes
0
answers
91
views
Semigroups of nondecreasing functions
Consider some partially ordered set $(E,\leq)$. Assume either that it is countable with the discrete topology, or that it has some topology compatible with the order, preferably one that makes it into ...
4
votes
1
answer
312
views
Applications of semigroups
I am limitedly aware of some works done in connecting semigroups and computer sciences. And, indeed, there are many works in this area.
May I ask you what would the ingeniously work be with this ...
- 233
10
votes
1
answer
355
views
Is Zariski closure of finitely generated matrix semigroup computable?
In general, can the Zariski closure of the semigroup of matrices $\langle M_1, \ldots, M_k \rangle$ be algorithmically computed (at least in theory)?
For this purpose I'm happy to assume the ...
- 163
2
votes
0
answers
98
views
If $H$ is an atomic, unit-cancellative monoid such that the set of atoms of $H$ is finite up to associates, then $H$ is BF
In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. ...
- 10.5k
1
vote
1
answer
166
views
Reference for a proof of cancellation property of braid monoids
Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.
...
- 6,201
5
votes
0
answers
64
views
Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
- 289
5
votes
0
answers
395
views
Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
4
votes
1
answer
434
views
Regarding a new algebraic structure
By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this )
$$
x*(y\cdot z)=x*y*z\;\; ; \;...
- 995
4
votes
1
answer
154
views
Right reversibility of submonoids of nilpotent groups
Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$).
A (cancellative) monoid/semigroup $S$ is right reversible if for ...
- 4,679
7
votes
1
answer
167
views
For which abelian groups $G$ does the monoid of zero-sum sequences over $G$ embed into a ring as a divisor-closed subsemigroup?
Let $K$ be a multiplicatively written semigroup (either commutative or not) and $H$ a subsemigroup of $K$. We say that $H$ is divisor-closed (in $K$) if $x \in H$ for all $x, y \in K$ such that $x \...
- 10.5k
21
votes
1
answer
759
views
Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure
Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...
- 13.6k
3
votes
0
answers
47
views
Counting the monic atoms $f$ in the semiring $\mathbf N[x]$ with $f(0)=1$, bounded coefficients, and degree $k$ (in the limit as $k \to \infty$)
Let $H$ be the multiplicative monoid of the (usual) semiring of polynomials in one variable $x$ with coefficients in $\mathbf N$. Given $\alpha, k \in \mathbf N$, denote by $\mathcal A_k(\alpha)$ the ...
- 10.5k
9
votes
1
answer
193
views
Detecting/Characterising positive elements in free groups
Let $X$ be a set, and let $F(X)$ be the free group generated by $X$.
I will say that an element of $F(X)$ is positive if it is in the monoid generated by all the conjugates in $F(X)$ of every member ...
- 2,178
3
votes
1
answer
106
views
Representation-finite symmetric monoid algebras
Is it known when a monoid algebra over a field is symmetric and representation-finite? For groups the answer is nice, so maybe there is a nice generalisation giving conditions on the field and the ...
- 26.5k
2
votes
1
answer
220
views
Alternative notation for Kleene star
I am writing a paper which use two different operations on sets of works $X$, both of which I want to denote by a star, $X^{\ast}$. One of these operations is the Kleene star, and for whatever reason ...
- 21
4
votes
1
answer
173
views
On the factorization of powers of atoms in the ring of integers of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is a non-unit element $a \in H$ that doesn't split into the product of two non-unit elements.
Given $x \in H$, we ...
- 10.5k
11
votes
2
answers
574
views
Identifying a group without 2-torsion
Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.) Is there ...
- 18.7k
5
votes
2
answers
252
views
Monoid of continuous self-maps of (real) surfaces
Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi_{1}(S),\pi_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of ...
- 6,995
9
votes
4
answers
1k
views
When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)
The title has it all. I'm looking for a reference to the following:
Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
- 10.5k
5
votes
1
answer
334
views
Short proof a monoid is a group iff every splitting is right homogeneous
In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...
- 10.5k
1
vote
1
answer
317
views
Monoids (or semigroups) with a "finite decomposition" property
In my research I have come across the following condition on a monoid.
Every element $x$ satisfies the following property: there exists a natural number $n$ such that for any $m \geq n$ and any ...
- 408
1
vote
1
answer
96
views
If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ ...
- 10.5k
4
votes
1
answer
385
views
Which monoids can be realized as the monoid of ideals of a commutative monoid?
Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an ideal of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) ...
- 10.5k
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
4
votes
1
answer
364
views
Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
- 10.5k
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
9
votes
1
answer
154
views
Inductive and reducible functions
The question was asked by a Computer Scientist and is closely related to parallel computing. But it is clearly of algebraic nature, so I decided to post it here.
Let $X$ be a set and $\bar X$ be the ...
user6976
2
votes
0
answers
62
views
Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
- 10.5k
8
votes
2
answers
352
views
Non-density of invertible elements in $\ell_1(\mathbb{N}_0)$
Consider the Banach algebra $\ell_1(\mathbb{N}_0)$ (with convolution / Cauchy product of series). I am looking for an elementary proof of the fact that the group of invertible elements in this algebra ...
- 11.3k
3
votes
1
answer
140
views
Monoids where every two non-unit elements have a common power
Q1. Is there any standard name for a (multiplicatively written) monoid $H$ with the property that, for all $x, y \in H \setminus H^\times$, there exist $m, n \in \mathbf N^+$ and $u, v \in H^\times$ ...
- 10.5k
6
votes
2
answers
422
views
Monoids in which every prime is an atom
Let $H$ be a multiplicatively written monoid with identity $1_H$. We write $H^\times$ for the set of units (or invertible elements) of $H$. We say that an element $a \in H$ is an atom if $a \notin H^\...
- 10.5k
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
9
votes
2
answers
1k
views
Ternary associative multiplication
In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such ...
- 4,780
8
votes
1
answer
314
views
What algebras are quotients of $\ell_1(\mathbf{N})$?
Every separable Banach space is a linear quotient of $\ell_1$, however not every separable Banach algebra is a Banach-algebra quotient of $\ell_1(G)$ for some group $G$ (these are the so called ...
- 331
1
vote
0
answers
58
views
A class of finitely generated semigroups
Let $G$ be a finitely generated group with a probability measure $\nu$. Suppose we have a finite first moment function $f:G\to \mathbb{R}$, i.e, such that $\Sigma_{g\in G}f(g)\nu(g)< \infty$. Then, ...
- 49
3
votes
0
answers
110
views
Distributive lattices -> left regular bands -> Atomistic lower semimodular lattices
Consider the following construction : let $(L,\vee,\wedge)$ be a finite distributive lattice, and let $(\mathrm{Int}(L),\star)$ be the monoid defined on the set of non empty intervals of $L$
$$\mathrm{...
- 2,751
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
5
votes
2
answers
137
views
The smallest number of vertices for a graph with the same endomorphism monoid
Let $G$ be a directed graph without loop and suppose that $M$ is its endomorphism monoid.
First how we can create a simple graph $\Gamma$ (using $G$), with the same endomorphism monoid? and then how ...
- 237
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
5
votes
0
answers
99
views
Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
5
votes
1
answer
243
views
Terminology for a monoid $H$ s.t. $xy \in H^\times$ only if $x, y \in H^\times$
The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of ...
- 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
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