Questions tagged [semigroups-and-monoids]

A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). A monoid is a semigroup with a neutral element. Of course, any group is also a monoid/semigroup.

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77 views

Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?

Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as $$ \mathcal{U} * \mathcal{V} = \left\{ A \...
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1answer
95 views

Cohomology of commutative monoid acting on module

I have a some naive questions about how to define the cohomology of a commutative monoid. One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $\text{Ext}^i_{\...
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1answer
281 views

Are there overwhelmingly more finite monoids than finite spaces? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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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 ...
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Free monoids on posets

I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying if (but not only if) $s_1, s_2, \ldots, s_r, t_1, ...
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58 views

When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open. Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace. Under what ...
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1answer
409 views

Is the Petersen graph a “Cayley graph” of some more general group-like structure?

The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?
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1answer
123 views

Lax monoidal functor

Let me denote $Cat$ the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let $F: (Cat, \times)\rightarrow (Set,\times)$ a symmetric monoidal ...
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1answer
631 views

Are there any “simple” monoids with intermediate growth?

The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...
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Size of the kernel (minimal ideal) of a finite semigroup

Let $A$ be an irreducible nonnegative $N\times N$ integer matrix with constant row sum $D$. Let $A_1, \dots, A_D$ be nonnegative integer matrices, each with constant row sum $1$, such that $\sum_k A_k ...
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143 views

The forgetful functor from Groups to Semigroups

While teaching this term I found myself reminded of the fact that the "usual" definition of a group homomorphism is really the definition of a semigroup homomorphism, applied to semigroups ...
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Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?

Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
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1answer
75 views

Derivable relations in a monoid

Let $ X $ be a monoid which is generated by the elements $ x_1, x_2, \hat x_1, \hat x_2 $ and the relations $ \hat x_i x_i = 1 $ and $ x_i \hat x_j = \hat x_j x_i $ for any distinct $ i, j = 1, 2 $. ...
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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[...
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Cuntz semigroups of basic C*-algebras

I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103): "[...] $A_i$ is ...
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Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example?

Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example? If a ring has an involution f, then f is an anti-automorphism;...
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57 views

Monoids with three or more “natural” partial orders

For any given monoid $M$ there may exist lots and lots of compatible pre-orders $\leq$. Only few of these are usually any interesting though. I can find some examples of monoids that have two non-...
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1answer
106 views

Well-foundedness of divisibility vs well-foundedness of right- and left-divisibility

Say that a preorder (i.e., a reflexive and transitive binary relation) $\preceq$ on a set $X$ is artinian if there is no sequence $(x_n)_{n \ge 1}$ of elements of $X$ with $x_{n+1} \prec x_n$ for ...
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Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?

Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
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1answer
294 views

Computations of divisor class monoids

Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors". Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
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1answer
75 views

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 ...
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3answers
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Is each squared finite group trivial?

A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective. Problem: Is each squared finite group ...
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75 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 ...
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43 views

Cayley's theorem for regular semigroups

"Cayley's theorem" for semigroups says that every semigroup of size $n$ is isomorphic to a subsemigroup of the semigroup of transformations $T_n$ or $T_{n+1}$. For inverse semigroups, we ...
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1answer
152 views

Conjugacy classes of monoids II: Abelianising a monoid, wrongly

$\newcommand{\unsim}{\mathord{\sim}}$Let $G$ be a group. What is $$ G/\left(ab\sim ba\ \middle|\ a,b\in G\right)? $$ Answer: not $G^{\mathrm{ab}}$, but the set of conjugacy classes of $G$. When ...
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68 views

What is known about the algebraic completion of a monoid?

It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid: Let $W$ be a monoid and let $p(x)=q(...
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Have you ever seen a partial binary operation that is not the morphisms of a semicategory, but it is covered by subsets that are?

Does the definition of a partial binary operation covered by its closed morphic (see below) subsets appear in the literature? Does this example? For the definition of "closed", see [1]. ...
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2answers
246 views

Extending monoids to a ring

I started reading about monoids (and semigroups in general) and came across of the example of some non-commutative monoids which cannot be endowed with some addition turning it into a ring (the monoid ...
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222 views

Has an “algebraic manifold” been defined before? Are there any non-trivial examples?

Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-closed if the following condition holds: for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $...
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231 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] ...
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337 views

Cancellation property for commutative monoid

Let $(M,+,e)$ be a commutative monoid with unit $e$. An element $a\in M$ is called cancellative element if for any $b,c \in M$ such that $a+b=a+c$ implies that $b=c$. Let $(\mathbf{N},+,0)$ the ...
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71 views

What is the real name for the initial object in the category of “monoid-valued measures of intervals” on transitive relations?

(I'm not asking for a true/false answer; I have a true algebraic fact and I'm looking for a reference in the literature. By the way, there is a version of this theorem that replaces monoid with $R$-...
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46 views

Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?

The terms are defined in a related question. [1] Conjecture 1. Let $A$ be a set, $W$ a cancellative invertible-free monoid, and $\cdot\colon A\times W\rightarrow A$ a cyclic right $W$-action generated ...
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1answer
127 views

Is every invertible-free cancellative monoid action represented by “shifting” certain maps?

[Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments] Let $W,X$ be ...
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2answers
149 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.
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terminology for a kind of two-sided module over a monoid

If $M$ is a monoid object in a pointed category $\mathcal{C}$, then a right $M$-module is an object $X$ equipped with a morphism $\alpha: X\times M\to X$ that satisfies the usual rules. There are ...
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1answer
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Generalized cancelation properties ensuring a monoid embeds into a group

Context: an obvious necessary condition for a monoid to embed into a group (as submonoid) is to satisfy the left and right cancelation rules: $$xy=xz \quad\Longrightarrow y=z;$$ $$yx=zx \quad\...
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1answer
327 views

Can every cancellative invertible-free monoid be embedded in a group?

A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$. Question: Can every cancellative invertible-free monoid be embedded in a group? I'm fairly sure that a quotient of the free product ...
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1answer
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Are cofibrations in topological monoids preserved by forming the product with the identity?

Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...
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2answers
181 views

“Completion property” in $(\beta\omega,+)$

Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, ...
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1answer
191 views

Addition and Rudin-Keisler ordering in $\beta \omega$

$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends ...
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243 views

On logarithmic schemes

I have two questions on logarithmic schemes Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
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204 views

Does every finite poset have a rigid endomorphism?

Crossposted on Mathematics. In this post, an order-preserving self-map of a poset $X$ will be called an endomorphism of $X$, and such an endomorphism $f$ will be called rigid if the only automorphism ...
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86 views

Heuristics for the word problem for monoids

The question is about a purely practical problem: Given is a list of identities, as in http://www.findstat.org/MapsDatabase/Mp00069: ...
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32 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(...
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1answer
181 views

Extending a monoid object in a category

A monoid object in a pointed category $\mathcal{C}$ is an object $M$ equipped with a multiplication morphism $\mu: M\times M\to M$ that is associative and unital, meaning that the diagrams that ...
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107 views

Classifying spaces of amalgamated topological monoids

Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
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1answer
116 views

On the width of the Catalan monoid and the rank of K-groups of the Furstenberg transformation group

The semigroup algebra of the Catalan monoid is isomorphic to the incidence algebra of $P_n$, where $P_n$ is the poset consisting of subsets of { 1,...,n } where for two subsets $X \leq Y$ if and only ...
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1answer
363 views

Does every set have a rigid self-map?

The question was asked on Mathematics Stackexchange but has remained unanswered so far. A self-map is a map $f:X\to X$ from a set $X$ to itself. There is an obvious notion of morphism, and thus of ...
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75 views

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 ...

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