All Questions
Tagged with or semigroups-and-monoids semigroups-and-monoids
606 questions
2
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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
2
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0
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73
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Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
2
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0
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145
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Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 2,605
2
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0
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119
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The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
2
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203
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Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
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0
answers
74
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Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.5k
2
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89
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Semigroups associated to binary necklaces and their semigroup algebra
I came across the following semi-group and the associated finite dimensional semi-group algebras over a field $K$ (which are Nakayama algebras) as they have very nice homological properties. My ...
- 26.5k
2
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0
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96
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Non-commutative version of the order dimension of a poset
I view the order dimension of a poset $P$ as an inherently commutative notion. On the one hand, it can be defined via realizers, which I find fairly intuitive from an order-theoretic viewpoint. On the ...
- 51
2
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0
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116
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Bijection between monoid of nilpotent decreasing self-maps and local subsemigroup
Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($...
- 355
2
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0
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33
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On the number of connected functional digraphs recoverable from the preimage set size structure
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,
$P_j=\left[f^{-j}(...
- 23
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91
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Is the natural action of the monoid of endomorphisms is a complete invariant for group?
Let $\alpha$ and $\beta$ be actions of semigroups $A$ and $B$ on sets $X$ and $Y$ respectively. Recall that $\alpha$ and $\beta$ are called isomorphic if there exists an isomorphism $\phi$ between ...
- 6,962
2
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48
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Compute irreducibles of monoid
Given $n > 0$ and $w \in \mathbb{Z}^n$. Is there an efficient algorithm to compute the set of irreducible elements of the monoid $M_w = \{x \in \mathbb{N}^n \mid \langle x,w\rangle = 0 \}$?
Here, ...
2
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60
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Are there finitely-presented astral monoids?
We say a semigroup $S$ is $k$-astral if there exists a finite set $F \subset S$ such that
whenever $s_1, s_2, ..., s_k \in S$ there exists $s \in S$ such that $\forall i: s_i \in sF$. Say $S$ is ...
- 6,652
2
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80
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Are the roots of an infinitely divisible probability infinitely divisible themselves?
Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too?
A sufficient criterion would be to ...
- 5,407
2
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81
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A semigroup property related to von Neumann regularity
A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...
- 18.7k
2
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50
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Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...
- 617
2
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0
answers
169
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What is the difference between a monosemiring and a semigroup?
What is the difference between a monosemiring and a semigroup?
The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity ...
- 203
2
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answers
63
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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
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0
answers
91
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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 ...
2
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0
answers
98
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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
2
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answers
62
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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
2
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0
answers
51
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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
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0
answers
106
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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
2
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0
answers
178
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Monoid prime ideals and prime congruences
I was wondering what the connection is between the notion of "prime congruence" on a monoid, and the notion of "prime ideal" in a monoid. Starting from a prime ideal $P$ in a monoid $M$, one can ...
- 4,547
2
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0
answers
87
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Terminology for torsion semigroups where the order of elements is uniformly finite
A (multiplicatively written) semigroup $\mathbb A = (A, \cdot)$ with the property that ${\rm ord}_\mathbb{A}(a) := |\{a^n: n \in \mathbf N^+\}| < \infty$ for every $a \in A$ is called a periodic (...
- 10.5k
2
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203
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Profinite Topology
Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism ...
- 421
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180
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Pro-p topology on free group
Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
- 421
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0
answers
139
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Goldie's Theorem for Semigroups
Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The ...
- 6,870
2
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216
views
Standard name for a Monoid/Semigroup with $a+b \leq a, b$?
I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice?
For instance, for reals $a,b > 0$, define $$a \oplus b = \frac{1}{\frac{1}{a}...
- 121
2
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0
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122
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First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
- 10.5k
2
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0
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124
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Reasoning about "approximately" associative structures and "almost monoids".
If $(M,+)$ is a monoid then it obeys the laws:
$$m_1 + 0 = 0 + m_1 = m_1$$
$$m_1+(m_2+m_3)=(m_1+m_2)+m_3$$
But what if I have a structure $(A,+)$ that is almost a monoid, but not quite. For example,...
- 21
2
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0
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272
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Continuity of multiplicative character
Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
- 61
1
vote
1
answer
190
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Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
- 87
1
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2
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367
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"Exactness" of groupify functor
For each commutative monoid $M$, there exists a "groupification" $\widehat{M}$, i.e. an abelian group that satisfies an obvious universal property.
I tried to prove the following: If in the diagram ...
- 9,853
1
vote
2
answers
296
views
Are orbits of an affine algebraic monoid affine?
Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...
- 4,902
1
vote
3
answers
585
views
Terminology for certain monoids which are to monoids like fields are to rings
Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "...
1
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2
answers
375
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What are the monoids in which every globally idempotent subsemigroup contains the identity element?
A semigroup is called globally idempotent when for any $x\in S$ there are $y,z\in S$ such that $x=yz$.
Is there a name for monoids whose every globally idempotent subsemigroup contains the identity ...
- 1,445
1
vote
2
answers
442
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submonoids of Z_n
Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative
monoid?
1
vote
1
answer
215
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Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 313
1
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2
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339
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Are monoids with zero and partial homomorphisms related?
Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U \...
- 2,464
1
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1
answer
231
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Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,405
1
vote
2
answers
378
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Is this a pre-ordered commutative semigroup?
Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
- 33.3k
1
vote
1
answer
106
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Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
1
vote
1
answer
142
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Congruences that aren't "finite from above," take 2: semigroups
This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
- 20.9k
1
vote
1
answer
90
views
Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 1,374
1
vote
1
answer
166
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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
1
vote
1
answer
400
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Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 1,045
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
- 1,882
1
vote
2
answers
394
views
Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 4,636
1
vote
1
answer
77
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What is the minimal possible size of a subset of this semigroup satisfying the following conditions?
Suppose $A$ is some set. Let's define a pair semigroup over $A$ as $P[A] = (A\times A \cup \{0\}, \circ)$, where the $\circ$ operation is defined by the following two identities:
$\forall a \in P[A]$ ...
- 2,618