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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Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
Tom Copeland's user avatar
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28 votes
1 answer
1k views

Is the Golomb countable connected space topologically rigid?

The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
Taras Banakh's user avatar
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5 votes
1 answer
236 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 ...
Salvo Tringali's user avatar
27 votes
13 answers
4k views

Homological algebra for commutative monoids?

Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
Chris Schommer-Pries's user avatar
2 votes
1 answer
208 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\...
Salvo Tringali's user avatar
39 votes
5 answers
4k views

What are the main structure theorems on finitely generated commutative monoids?

I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...
John Baez's user avatar
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25 votes
2 answers
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The number of polynomials on a finite group

A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
Taras Banakh's user avatar
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13 votes
1 answer
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For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?

It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\...
cha21's user avatar
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4 answers
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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 ...
Salvo Tringali's user avatar
5 votes
1 answer
540 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 ...
David Pokorny's user avatar
13 votes
3 answers
966 views

Model Structure/Homotopy Pushouts in topological monoids?

Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$. Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
Joey Hirsh's user avatar
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7 votes
1 answer
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Pushouts of injective monoid homomorphisms

Given a pushout square in the category of monoids $$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and ...
HeinrichD's user avatar
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7 votes
2 answers
506 views

A linearly orderable monoid which does not embed into a linearly orderable group

It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable ...
Salvo Tringali's user avatar
7 votes
3 answers
515 views

Is the class of inverse semigroups globally determined?

This question is a follow-up to this one I asked on math.stackexchange. I've decided to ask here because I believe this is a research-level question. I'm sorry if I'm wrong -- I'm not a researcher ...
Michał Masny's user avatar
4 votes
1 answer
161 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 ...
David Pokorny's user avatar
4 votes
1 answer
348 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 $M := H \setminus H^\times$,...
Salvo Tringali's user avatar
3 votes
1 answer
191 views

Centralizer of a single element in the monoid of self-maps of a set

This is a follow-up to this question: For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both? Let $X$ be a set, and $X^...
YCor's user avatar
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3 votes
2 answers
160 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 $$XY := \{xy: x \in X,\, y \in Y\}.$$ ...
Salvo Tringali's user avatar
3 votes
1 answer
120 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 \...
Salvo Tringali's user avatar
1 vote
0 answers
247 views

partially commutative monoid [closed]

Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
GA316's user avatar
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1 vote
1 answer
139 views

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 ...
Noah Schweber's user avatar
49 votes
3 answers
2k views

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 ...
Taras Banakh's user avatar
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41 votes
4 answers
2k views

What is the probability two random maps on n symbols commute?

It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
Benjamin Steinberg's user avatar
33 votes
0 answers
2k views

Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?

For a (discrete) monoid $M$, the classifying space $BM$ is the geometric realization of the nerve of the one object category whose hom-set is $M$. (This definition gives the usual classfiying space ...
Omar Antolín-Camarena's user avatar
31 votes
5 answers
8k views

How many binary operations are associative?

Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each be ...
Joseph O'Rourke's user avatar
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
24 votes
3 answers
2k views

Non-abelian Grothendieck group

By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{...
Martin Brandenburg's user avatar
22 votes
5 answers
2k views

Lie groups vs Lie monoids

Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
Benjamin's user avatar
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16 votes
2 answers
1k views

The symmetric monoidal category of finite sets

It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...
Martin Brandenburg's user avatar
15 votes
2 answers
1k views

Exact sequence of monoids

What is the right definition of an exact sequence of monoid homomorphisms? I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6, http://www.math.ucla.edu/~balmer/Pubfile/...
John Voight's user avatar
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15 votes
1 answer
2k views

Automorphisms of $P(\Bbb N)$

I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the ...
Michał Masny's user avatar
14 votes
2 answers
973 views

Recovering a monoidal category from its category of monoids

What kind of additional properties and/or structures one needs to impose on the category of (commutative or noncommutative) monoids of some monoidal category so that one can recover the original ...
Dmitri Pavlov's user avatar
13 votes
2 answers
1k views

Economical hard word problem

Can anyone give me an example of a very simple word problem, where by "simple" I mean that it has very few generators and relations, that is nevertheless insoluble. To make the question easier, I am ...
gowers's user avatar
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12 votes
1 answer
723 views

Is the following construction of the 0-Hecke monoid (well) known?

Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
Benjamin Steinberg's user avatar
11 votes
3 answers
905 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
Thomas Klimpel's user avatar
11 votes
3 answers
993 views

The concept "conjugate class" in monoids.

Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.
Jianrong Li's user avatar
  • 6,101
11 votes
2 answers
541 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 ...
Pace Nielsen's user avatar
10 votes
1 answer
660 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
10 votes
2 answers
668 views

On functors preserving monoid objects

If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids: ...
LorenzoPerticone's user avatar
10 votes
1 answer
262 views

A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids

In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following. ...
Benjamin Steinberg's user avatar
10 votes
0 answers
306 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
Jakub Konieczny's user avatar
10 votes
1 answer
1k 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
10 votes
1 answer
431 views

Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
Dmitri Pavlov's user avatar
10 votes
1 answer
402 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 ...
Pierre-Yves Gaillard's user avatar
9 votes
5 answers
1k views

On the notion of partial semigroup

A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = (M,...
Salvo Tringali's user avatar
9 votes
2 answers
2k views

What is the free monoidal category generated by a monoid?

In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category $...
ziggurism's user avatar
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8 votes
0 answers
276 views

Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question: ...
wlad's user avatar
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8 votes
3 answers
1k views

Structure Theorem for finitely generated commutative cancellative monoids?

Is there a Structure Theorem for finitely generated commutative cancellative monoids? Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...
Martin Brandenburg's user avatar
8 votes
5 answers
1k views

Examples of left reversible semigroups

I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See this wikipedia page for the definition of a left ...
Orr Shalit's user avatar
8 votes
1 answer
233 views

Functions over monoids which factor in two different ways

This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there. Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
Tony Huynh's user avatar
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