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2 votes
0 answers
116 views

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 ($...
1Spectre1's user avatar
  • 355
25 votes
3 answers
1k views

What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?

This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
Dominic van der Zypen's user avatar
6 votes
3 answers
472 views

Spaces with unique endomorphism monoids

If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$. We ...
Dominic van der Zypen's user avatar
3 votes
1 answer
125 views

Quasi-isometries and E-unitary inverse semigroups

Let $S = \langle K\rangle$ be a finitely generated inverse semigroup, where $K \subset S$ is a fixed, finite and symmetric set of generators. Preliminaries: Recall that we say that $s, t \in S$ are $\...
Diego Martinez's user avatar
6 votes
1 answer
128 views

Ascending sequences of idempotents in inverse semigroups

I've enocuntered the following question in my current research, and I'd appreciate any help you could give me. This is probably well known to experts on the subject. Let $S = \langle K \rangle$ be a ...
Diego Martinez's user avatar
2 votes
0 answers
33 views

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}(...
bmf's user avatar
  • 23
2 votes
1 answer
156 views

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{...
Surajit's user avatar
  • 73
4 votes
1 answer
446 views

What is a "cusp" ("кусок") in relation to Guba's embedding theorem?

I'm confused by the definition of a "cusp" as found in V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link). In the words of Mark ...
Salvo Tringali's user avatar
2 votes
0 answers
91 views

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 ...
Arshak Aivazian's user avatar
3 votes
1 answer
252 views

Classification of associative polynomial functions

What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers ...
Arshak Aivazian's user avatar
14 votes
1 answer
792 views

Associativity may fail by little?

It is a well-known result on group theory that if a group has many pairs of commuting elements then it is abelian. This motivated the following pseudo-conjecture. If a (possibly infinite) set $S$ ...
Luis Ferroni's user avatar
  • 1,889
1 vote
1 answer
326 views

Closed submonoid of $(\mathbb{C}^*)^n$

The answer of this question might be known but I was not able to find any answer. Let $n\geq 1$ and $S$ be a closed submonoid of $(\mathbb{C}^*)^n$, that is, a closed and stable by product subset of $(...
phdstud's user avatar
  • 143
4 votes
1 answer
108 views

Closed cobounded additive submonoid of $\mathbb{R}^n$

Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more ...
phdstud's user avatar
  • 143
1 vote
1 answer
77 views

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: $\forall a \in P[A]$ ...
Chain Markov's user avatar
  • 2,618
7 votes
2 answers
383 views

Counting nilpotent self-maps of $\{1,\dots,n\}$ with image of a given cardinal

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(...
1Spectre1's user avatar
  • 355
6 votes
1 answer
449 views

Comonoids in the category of monoids

Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids? ...
HeinrichD's user avatar
  • 5,482
2 votes
0 answers
48 views

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, ...
Kasper Dokter's user avatar
1 vote
1 answer
231 views

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 ...
ABIM's user avatar
  • 5,405
18 votes
2 answers
1k views

Monoids of endomorphisms of nonisomorphic groups

Can monoids of endomorphisms of nonisomorphic groups be isomorphic ?
Arshak Aivazian's user avatar
1 vote
1 answer
163 views

Internal commutative monoid gives commutative monad

Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object. The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
geodude's user avatar
  • 2,129
4 votes
1 answer
134 views

Joint spectral radius of $\{M,M^T\}$

Let $F$ be a bounded subset of ${\bf M}_n({\mathbb C})$. G.-C. Rota & G. Strang defined the joint spectral radius of $F$ as follows. For $k\ge1$, denote $F_k$ the set of all products of $k$ ...
Denis Serre's user avatar
  • 52.3k
3 votes
0 answers
83 views

Cancellativity of a particular $2$-generated monoid presented by an infinite number of relations

Let $X = \{x, y\}$ be a two-element set, and let $H$ be the monoid defined by the presentation $$ \langle x, y \mid x y^k x = y x y^{k+1} x y, \text{ for } k = 0, 1, 2, \ldots\rangle. $$ That is, $H$ ...
Salvo Tringali's user avatar
2 votes
0 answers
60 views

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 ...
Ville Salo's user avatar
  • 6,652
7 votes
0 answers
260 views

Generating the monoid of injective endomorphisms of the free group

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...
Lvzhou Chen's user avatar
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)\...
gete's user avatar
  • 203
0 votes
1 answer
296 views

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\...
Adi Ostrov's user avatar
6 votes
0 answers
132 views

Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
Joshua Meyers's user avatar
8 votes
2 answers
585 views

Is the equational theory of groups axiomatized by the associative law?

Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory ...
user107952's user avatar
  • 2,023
5 votes
1 answer
303 views

A characterisation of faces of rational polyhedral cones

This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
Fred Rohrer's user avatar
  • 6,700
5 votes
1 answer
170 views

Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?

McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents. Question: Which homotopy types ...
Tim Campion's user avatar
  • 63.9k
10 votes
1 answer
2k views

Who invented Monoid?

I was trying to find (and failed) the original author of either the concept of Monoid (set with binary associative operation and identity) the name (which sounds french ? and also Dioid (for what ...
c69's user avatar
  • 203
5 votes
1 answer
205 views

Topological category of topological monoids / operads

The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...
Keke Zhang's user avatar
4 votes
1 answer
307 views

Characterization of Archimedean linearly ordered monoids

In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is ...
eyeballfrog's user avatar
6 votes
0 answers
47 views

Special monomorphism to encode the inclusion of topological submonoids

Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms. Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with ...
geodude's user avatar
  • 2,129
1 vote
0 answers
69 views

How exactly to adapt Brown's collapse from monoids to algebras?

In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly: Given a simplicial set $X$ equipped with a ...
Hilario Fernandes's user avatar
0 votes
1 answer
84 views

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 ...
bmf's user avatar
  • 23
2 votes
1 answer
195 views

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},\...
1Spectre1's user avatar
  • 355
1 vote
0 answers
202 views

What is the normalized complex of a simplicial set with a monoid action?

This question is a follow up to this question I posted on Math.SE. I will make this question self-contained, though. In a certain point on the paper The Geometry of Rewriting Systems, Kenneth Brown ...
Hilario Fernandes's user avatar
3 votes
1 answer
244 views

Category of continuous self maps

Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)? How can we tell whether a category is the category of continuous ...
alesia's user avatar
  • 2,772
1 vote
0 answers
132 views

Is the Upper Banach density always zero with respect to some sequence of Finite subset

The following question came to me while reading the paper 'Density in Arbitrary Semigroups' by Hindman and Strauss. Question: Given an infinite subset $A$ of $\mathbb{N}$ such that $A^c$ is also ...
Surajit's user avatar
  • 73
3 votes
1 answer
62 views

Is a J-simple semigroup with an idempotent necessarily regular?

If a semigroup S has no proper ideals can it have both regular and non-regular members? My guess would be 'yes' but in that case does anyone know of an example in the literature?
Peter Higgins's user avatar
1 vote
0 answers
55 views

Schemes for conditional distributions (monads)

(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.) Suppose you have a monad ...
prdnr's user avatar
  • 121
-3 votes
1 answer
234 views

A common name for a functorial construction of Commutative Algebra?

I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name. Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
Taras Banakh's user avatar
  • 41.9k
1 vote
0 answers
111 views

When a semigroup ideal is a determinantal ideal?

Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
Paolo1994's user avatar
  • 113
5 votes
1 answer
254 views

Examples of Yang-Baxter monoids

Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities: $(X,\circ,1)$ is a monoid, $f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$ $x\circ y=f(x,y)\circ ...
Joseph Van Name's user avatar
2 votes
1 answer
229 views

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 $a,A,b$...
goblin GONE's user avatar
  • 3,793
2 votes
1 answer
176 views

Generating totally ordered free commutative monoids

Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$. When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am ...
Tartrate's user avatar
  • 341
1 vote
1 answer
70 views

Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations

It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...
Ethan Splaver's user avatar
1 vote
1 answer
126 views

Do these sorts of submonoids go by a particular name?

Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows: $$r(x)=\{y\in M:xy=x\}$$ $$l(x)=\{y\in M:yx=x\}$$ Do these sorts of sub-monoids go by a particular name?...
user3865391's user avatar
10 votes
1 answer
673 views

Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?

Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
Ethan Splaver's user avatar

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