All Questions
Tagged with or semigroups-and-monoids semigroups-and-monoids
606 questions
2
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1
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167
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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
2
votes
1
answer
182
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Terminology for the equation $a=a+b$ in commutative semigroups
Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often ...
2
votes
2
answers
278
views
How much information does the multiplicative semigroup of an algebra contain?
How much do we know about an given algebra when we only know its semigroup strucure under the product law?
How far can two algebras be distinguished by knowing only their semigroup strucure?
The ...
- 221
2
votes
1
answer
259
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Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
- 2,029
2
votes
1
answer
302
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Name for this algebraic structure?
I've found myself looking at a structure $\mathbb{M}$ whose important properties are:
$\mathbb{M}$ is a discretely ordered additive monoid.
$\mathbb{M}$ has a least element, and this least element is ...
- 10.1k
2
votes
1
answer
169
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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
2
votes
2
answers
180
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a question about semigroups
Let $S$ be a semigroup and $I,J$ be two ideals of $S$. For a semilattice we know that $IJ=I\cap J$. Now the question is there a semigroup with the property $IJ=I\cap J$. thanks for your attention
- 23
2
votes
3
answers
501
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Generalized free product of semigroups with amalgamated subsemigroups
Hanna Neumann in
[American Journal of Mathematics, 1948,
http://www.jstor.org/discover/10.2307/2372201?uid=2&uid=4&sid=21102497379451 ]
introduced a notion of generalized free product of ...
- 3,110
2
votes
1
answer
57
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Are simplicial commutative inverse semigroups fibrant?
Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice ...
- 742
2
votes
1
answer
271
views
Apropos of two groups being globally isomorphic iff they are isomorphic
Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced ...
- 10.5k
2
votes
1
answer
174
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Understanding the picture of monoidal space
Ogus in his slides https://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf presents the following picture of a monoidal space $\operatorname{Spec}(\mathbb{N} \longrightarrow \mathbb{C}[\mathbb{N}])...
- 165
2
votes
1
answer
94
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What are the n-ary subsemigroups of $\mathbb{N}$?
There is a well-known result about the subsemigroups of $\mathbb{N}$ stating that the additive subsemigroup generated by a (finite) set $A$ of $\mathbb{N}$ is cofinite in $\mathbb{N}$ if and only if $\...
- 98
2
votes
1
answer
156
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Additivity of the upper Banach density
The following notion of upper Banach density was defined (Definition 2.1(c)) by Hindman and Strauss in their paper 'Density in arbitrary semigroups':
Definition: Let $S$ be a semigroup, let $\mathcal{...
- 73
2
votes
1
answer
229
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Has the "semidirect monoid of a semiring" been considered anywhere?
Given a semiring $S$, we get a monoid $M(S)$ as follows:
The underlying set of $S$ is $S^2$
The identity element is $(0,1)$
The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
- 3,793
2
votes
1
answer
220
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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
2
votes
2
answers
241
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If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
- 10.5k
2
votes
2
answers
1k
views
Classification of rings between a PID and its field of fractions?
Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$.
Theorem: Every such ring $R$ is a ...
- 3,761
2
votes
2
answers
293
views
Equivalence relations in suplattices
I am wondering about generalisations of the concept of equivalence relations to suplattices.
Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices ...
- 2,442
2
votes
2
answers
357
views
Projective limit construction of a semigroup
Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. ...
- 417
2
votes
1
answer
360
views
On a property of subsemigroups
Let $H$ denote a subsemigroup of a semigroup $G$.
I'm interested in the following property:
$$\forall g\in G\exists h\in H:gh\in H.$$
This property is weaker than the property that $H$ is an ideal ...
- 21
2
votes
1
answer
301
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Is there existing terminology for this technical condition on semilattices?
Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of $S$...
- 25.8k
2
votes
1
answer
117
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Primal identity in matrix semigroup
Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product.
We call $s_1\cdots s_k$ an identity index if
$M_{s_1}M_{...
- 1,503
2
votes
1
answer
116
views
On the maximum elements of a numerical semigroup that have order between $n$ and $2n$
Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
- 183
2
votes
1
answer
67
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$E$-separated semigroups
Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is ...
- 41.9k
2
votes
1
answer
166
views
Submonoid of free monoid with certain properties
Let $N$ be a submonoid of a free monoid $M$ such that
$m_1nm_2\in N \Rightarrow m_1,m_2\in N$ for any $m_1,m_2\in M$ and $n\in N\setminus\{1\}$. $\quad\quad\quad\quad$ (C)
Do such submonoids ...
- 491
2
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1
answer
148
views
Terminology for a ring where every right cancellable element is cancellable
Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
- 10.5k
2
votes
1
answer
404
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Reference request: a cousin to the log semiring
Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
- 15.4k
2
votes
1
answer
153
views
Define a homomorphism of a set of graphs to its power set
Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is,
$G_1\cup G_2$
$=\langle V(G_1)\cup V(G_2), (E(G_1)\...
- 203
2
votes
1
answer
195
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A question about semigroup union
The semigroup of all order-preserving and decreasing transformations in full transformations semigroup $T_n$ is denoted $C_n$.
I consider the idempotent set $A=\{\begin{bmatrix}2\\1 \end{bmatrix},\...
- 355
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
user111492
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
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
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
2
votes
1
answer
294
views
Finitely generated ordered monoids and noetherian subsets
(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
- 6,700
2
votes
1
answer
223
views
pseudovarieties and profinite group : do * and g() commute?
Let $V$ and $W$ be pseudovarieties of finite monoids. We denote with $gV$ the pseudovariety of categories generated by $V$, and by $V*W$ the semidirect product of pseudovarieties $V$ and $W$.
Does ...
- 421
2
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1
answer
132
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Maximal sub-inverse semigroups of $M_n(\mathbb{C})$ and $M_n(F_p)$
An inverse semigroup $S$ is a semigroup in which every element $x \in S$ has a unique inverse $y \in S$ such that $x = xyx$ and $y = yxy$. Are there some references characterizing the maximal sub-...
- 6,201
2
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1
answer
290
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Idempotents in Green J classes
I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...
- 233
2
votes
1
answer
796
views
Commutative, idempotent partially ordered monoids
A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for commutativity). ...
- 2,442
2
votes
1
answer
1k
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monoid ring and some structure within it - how is it called?
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
- 1,626
2
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0
answers
92
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Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
2
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0
answers
80
views
An alternative definition for finitely generated (and principal) ideals in a semigroup
Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
- 10.5k
2
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0
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91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
2
votes
0
answers
176
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On the origin of power semigroups
Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty ...
- 10.5k
2
votes
0
answers
68
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Semigroups related to iterated orthogonal complement
Let $R\subset V\times V$ be a relation on a set $V$. For a subset $S\subset V$, define its orthogonal complement with respect to $R$
as
$$S^l:=\{ x: \forall y\in S\ \ (x,y)\in R\},\ \ S^r:=\{y: \...
- 513
2
votes
0
answers
64
views
A particular generalization of free partially commutative monoids
A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
- 21
2
votes
0
answers
73
views
What should I call a log scheme with free reduced monoids?
This is a terminology question about a class of log varieties.
Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
- 7,952
2
votes
0
answers
161
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Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
user491484
2
votes
0
answers
181
views
So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
2
votes
0
answers
67
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Type of numerical semigroups is not bounded when embedding dimension is $\geq 4$
I am currently studying numerical semigroups. I know that there is no upper bound for the type of a numerical semigroup with embedding dimension greater or equal than $4$. There is a famous example by ...
- 121
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 31