All Questions
Tagged with semigroup-theory or semigroups-and-monoids
605 questions
11
votes
3
answers
1k
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.
- 6,201
11
votes
3
answers
939
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 ...
- 2,464
11
votes
2
answers
574
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 ...
- 18.7k
11
votes
2
answers
950
views
Define Turing machine with algebraic concepts/structures
Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, in this way) and then their definition is formalized (for example, in this way).
Is it ...
user60665
11
votes
1
answer
949
views
Magma "actions" (or alternatively, "What is the Yoneda lemma for magmas?")
Arguably the most import thing about groups, semigroups and more generally categories, is that they can act on sets (or even collections of sets in the case of a category). This is the basis for all ...
- 2,392
11
votes
0
answers
427
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Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 591
11
votes
0
answers
286
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 ...
- 4,416
11
votes
0
answers
214
views
Is it decidable if a tree-presented semigroup contains an idempotent?
A semigroup presentation $\langle A | R\rangle$ is called tree-like if every relation has the form $ab=c$, $a,b,c$ are in $A$ and if two relations $ab=c, a'b'=c'$ belong to $R$, then $c=c'$ if and ...
user6976
10
votes
2
answers
1k
views
Connective spectra versus simplicial abelian groups - very basic question
Hello,
I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).
I guess that connective spectra have a model ...
- 5,562
10
votes
1
answer
422
views
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:
$$xy=xz \quad\Longrightarrow y=z;$$
$$yx=zx \quad\...
- 63.9k
10
votes
1
answer
2k
views
Who invented Monoid?
I was trying to find (and failed) the original author of either
the concept of Monoid (set with binary associative operation and identity)
the name (which sounds french ? and also Dioid (for what ...
- 203
10
votes
1
answer
409
views
Does every set have a rigid self-map?
The question was asked on Mathematics Stackexchange
but has remained unanswered so far.
A self-map is a map $f:X\to X$ from a set $X$ to itself. There is an obvious notion of morphism, and thus of ...
- 4,416
10
votes
2
answers
716
views
On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
- 613
10
votes
1
answer
579
views
Group completion of topological monoids
Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\...
- 1,252
10
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,...
- 10.5k
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 ...
- 2,506
10
votes
1
answer
274
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.
...
- 38.6k
10
votes
1
answer
440
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 ...
- 37.8k
10
votes
2
answers
752
views
Adding a formal inverse of an element to a free monoid
Let $FM_2=\langle a,b\rangle$ be the free monoid of rank 2. If we add a formal inverse to the word $aba$, we get the free group $F_2$ (because both $a$ and $b$ will have inverses).
Question: For ...
user6976
10
votes
1
answer
355
views
Is Zariski closure of finitely generated matrix semigroup computable?
In general, can the Zariski closure of the semigroup of matrices $\langle M_1, \ldots, M_k \rangle$ be algorithmically computed (at least in theory)?
For this purpose I'm happy to assume the ...
- 163
10
votes
1
answer
243
views
Can a semigroup with zero be globally isomorphic to a semigroup without zero?
This is not a great question for sure and it may even be trivial for all I know, but a couple of years ago, when I still thought I'd be a mathematician, I spent quite a lot of time thinking about it ...
- 1,445
10
votes
2
answers
444
views
Iterated sumset inequalities in cancellative semigroups
This question is motivated by the following well-known theorems:
Thm (Plünnecke): If $A$ is a finite nonempty subset of an abelian group, then for every $n$ we have $|A^n| \le \frac{|AA|^n}{|A|^n}|A|$...
- 8,688
10
votes
0
answers
248
views
What is the tiling semigroup for an einstein "hat" tiling?
My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
- 379
10
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0
answers
367
views
A formula for Frobenius number of certain numerical semigroups
The old formula for the Frobenius number of a numerical semigroup generated by two elements can be stated as follows: assume $\gcd\{a+1,b+1\}=1$, then the Frobenius number of $S= \left<a+1,b+1\...
- 30.5k
10
votes
0
answers
314
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 ...
- 1,914
9
votes
4
answers
1k
views
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 ...
- 10.5k
9
votes
2
answers
667
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Semi group of polynomials which all roots lie on the unit circle
Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $.
The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials.
With the standard multiplication, $X$...
- 356
9
votes
2
answers
1k
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Ternary associative multiplication
In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such ...
- 4,780
9
votes
3
answers
1k
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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, ...
- 63.1k
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 $...
- 1,446
9
votes
5
answers
1k
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References/literature for pushouts in category of commutative monoids? [ed. - amalgams]
This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid ...
- 25.8k
9
votes
1
answer
743
views
Why is choice needed in Ellis' Lemma?
Ellis Lemma on idempotent elements asserts that:
Lemma (Ellis). Every compact semigroup has an idempotent.
The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.
...
- 1,412
9
votes
1
answer
211
views
Reference for Kakutani result on power sum bases of symmetric functions
Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
- 528
9
votes
1
answer
193
views
Detecting/Characterising positive elements in free groups
Let $X$ be a set, and let $F(X)$ be the free group generated by $X$.
I will say that an element of $F(X)$ is positive if it is in the monoid generated by all the conjugates in $F(X)$ of every member ...
- 2,178
9
votes
1
answer
154
views
Inductive and reducible functions
The question was asked by a Computer Scientist and is closely related to parallel computing. But it is clearly of algebraic nature, so I decided to post it here.
Let $X$ be a set and $\bar X$ be the ...
user6976
9
votes
1
answer
889
views
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 ...
- 10.5k
9
votes
0
answers
164
views
Parallelizability of Lie monoids
A Lie monoid is a monoid, together with a structure of a smooth manifold (possibly with a boundary), such that the monoid multiplication is smooth.
If all left (or right) translations in a Lie monoid $...
- 61
9
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0
answers
347
views
What is the precise connection between logarithmic algebraic geometry and the field with one element?
Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
- 11.8k
9
votes
0
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373
views
Embedding $\beta\mathbb{N}$ into a product of Cantor sets
Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
- 297
8
votes
3
answers
1k
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Are all free monoids residually finite?
I cannot manage to prove that a free monoid with operation concatenation, and with at least two generators is residually finite. If there is just one generator, the free monoid $\{a\}^*$ is isomorphic ...
- 81
8
votes
4
answers
1k
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Haar Measure on Locally Compact Semigroups
I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure?
...
- 91
8
votes
3
answers
609
views
Nielsen-Schreier theorem for monoids
Let $S$ be a finitely generated free abelian semigroup (or monoid), and let $T \subset S$ be a sub-semigroup (sub-monoid). Does the Nielsen-Schreier theorem hold in this case, that is, will $S$ still ...
- 459
8
votes
1
answer
1k
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Give an example of monoid with property $m^2 = m^3$
Give an example of finitely generated, infinite monoid $M$ with property that for all $m \in M$ we've got $m^2 = m^3$.
This question comes from the problem I was given during algebraic languages ...
8
votes
2
answers
427
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Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF?
The title question is too vague so let me be specific.
Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ...
- 38.6k
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
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 ...
- 2,013
8
votes
1
answer
448
views
Can a Shelah semigroup be commutative?
A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$...
- 41.8k
8
votes
5
answers
1k
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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 ...
- 550
8
votes
1
answer
322
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Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
8
votes
1
answer
645
views
Isomorphic morphisms. A 27-morphism category
Two morphisms of category $\ \mathbf C\ $ are isomorphic to one
another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that ...
- 4,786