Skip to main content

All Questions

Filter by
Sorted by
Tagged with
3 votes
1 answer
140 views

Monoids where every two non-unit elements have a common power

Q1. Is there any standard name for a (multiplicatively written) monoid $H$ with the property that, for all $x, y \in H \setminus H^\times$, there exist $m, n \in \mathbf N^+$ and $u, v \in H^\times$ ...
Salvo Tringali's user avatar
3 votes
0 answers
89 views

Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$

Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
Salvo Tringali's user avatar
3 votes
0 answers
92 views

Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category

Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
varkor's user avatar
  • 10.7k
3 votes
0 answers
103 views

An isomorphism problem for semigroups of ideals

An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $...
Salvo Tringali's user avatar
3 votes
0 answers
81 views

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 ...
Sophie M's user avatar
  • 695
3 votes
0 answers
197 views

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 ...
Sambo's user avatar
  • 285
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
3 votes
0 answers
47 views

Counting the monic atoms $f$ in the semiring $\mathbf N[x]$ with $f(0)=1$, bounded coefficients, and degree $k$ (in the limit as $k \to \infty$)

Let $H$ be the multiplicative monoid of the (usual) semiring of polynomials in one variable $x$ with coefficients in $\mathbf N$. Given $\alpha, k \in \mathbf N$, denote by $\mathcal A_k(\alpha)$ the ...
Salvo Tringali's user avatar
3 votes
0 answers
110 views

Distributive lattices -> left regular bands -> Atomistic lower semimodular lattices

Consider the following construction : let $(L,\vee,\wedge)$ be a finite distributive lattice, and let $(\mathrm{Int}(L),\star)$ be the monoid defined on the set of non empty intervals of $L$ $$\mathrm{...
Olivier Bégassat's user avatar
2 votes
2 answers
286 views

Idempotent semigroups: Are they all residually finite?

As pointed out by Mark Sapir in his answer to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove ...
Salvo Tringali's user avatar
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
2 votes
2 answers
475 views

On the notion of torsion-freeness in semigroup theory

The following seems to be the "official" notion of torsion-freeness in the context of semigroups: TF1. A (multiplicatively written) semigroup $\mathfrak A$ is torsion-free if there do not ...
Salvo Tringali's user avatar
2 votes
1 answer
259 views

Nuclearity of certain semigroup crossed product C*-algebras

This question is related to this question link. Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
user5831's user avatar
  • 2,029
2 votes
2 answers
241 views

If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post). Basically following some ideas of W. Lawvere (but not his ...
Salvo Tringali's user avatar
2 votes
2 answers
293 views

Equivalence relations in suplattices

I am wondering about generalisations of the concept of equivalence relations to suplattices. Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices ...
The User's user avatar
  • 2,442
2 votes
1 answer
301 views

Is there existing terminology for this technical condition on semilattices?

Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of $S$...
Yemon Choi's user avatar
  • 25.8k
2 votes
1 answer
67 views

$E$-separated semigroups

Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$. Observe that $X$ is ...
Taras Banakh's user avatar
  • 41.9k
2 votes
1 answer
148 views

Terminology for a ring where every right cancellable element is cancellable

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
Salvo Tringali's user avatar
2 votes
1 answer
404 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 ...
Steve Huntsman's user avatar
2 votes
1 answer
290 views

Idempotents in Green J classes

I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular: A $\mathcal J$-class containing an idempotent is called regular. ...
Mikasa's user avatar
  • 233
2 votes
0 answers
80 views

An alternative definition for finitely generated (and principal) ideals in a semigroup

Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
Salvo Tringali's user avatar
2 votes
0 answers
91 views

A recursive description of the smallest divisor-closed subsemigroup containing a set

Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid). ...
Salvo Tringali's user avatar
2 votes
0 answers
176 views

On the origin of power semigroups

Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty ...
Salvo Tringali's user avatar
2 votes
0 answers
64 views

A particular generalization of free partially commutative monoids

A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
rotas's user avatar
  • 21
2 votes
0 answers
181 views

So many types of subwords! How are they called?

Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
Salvo Tringali's user avatar
2 votes
0 answers
91 views

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 ...
Vilhelm Agdur's user avatar
2 votes
0 answers
62 views

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) ...
Salvo Tringali's user avatar
2 votes
0 answers
87 views

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 (...
Salvo Tringali's user avatar
2 votes
0 answers
122 views

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 $d: X \times X \to [0,\infty]$ ...
Salvo Tringali's user avatar
2 votes
0 answers
124 views

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,...
Mike Izbicki's user avatar
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 ...
Grisha Taroyan's user avatar
1 vote
1 answer
166 views

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 $b=c$. ...
Jianrong Li's user avatar
  • 6,201
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
1 vote
1 answer
317 views

Monoids (or semigroups) with a "finite decomposition" property

In my research I have come across the following condition on a monoid. Every element $x$ satisfies the following property: there exists a natural number $n$ such that for any $m \geq n$ and any ...
Aleš Bizjak's user avatar
1 vote
0 answers
56 views

Effect on finite transformation semigroup under a particular modification of the generators

The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
Sophie M's user avatar
  • 695
1 vote
0 answers
83 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(...
David Pokorny's user avatar
1 vote
0 answers
102 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 $R$-...
David Pokorny's user avatar
1 vote
0 answers
52 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 ...
David Pokorny's user avatar
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
1 vote
0 answers
82 views

What is known about the cohomology of the matrix monoid?

When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object). Let $M_n(R)$ be the monoid of matrices with matrix ...
Cihan's user avatar
  • 1,726
1 vote
0 answers
639 views

What is the real name of this relation and operation on a particular set of maps between cancellative monoids?

Let $A,B$ be cancellative monoids and define a transducer as a map $f\colon A \rightarrow B$ such that $f(1)=1$ and for all $a_1 ,a_2 \in A$, there exists a $b \in B$ such that $f(a_1 a_2)=f(a_1) b$. ...
David Pokorny's user avatar
1 vote
0 answers
99 views

Name for condition on map of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that $k(\epsilon)=\epsilon$ for all $a,b\in M$, there exists $v\in N$ such that ...
David Pokorny's user avatar
0 votes
2 answers
283 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.
Learner's user avatar
  • 141
0 votes
1 answer
232 views

Kernel elements for the Grothendieck group map of a commutative monoid

This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
Tom LaGatta's user avatar
  • 8,512
0 votes
1 answer
143 views

Left syndeticity and right syndeticity in nilpotent group

$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
Surajit's user avatar
  • 73
0 votes
1 answer
63 views

Monoid morphisms satisfying a decomposition condition

Let $A$ and $B$ be monoids, let $f\colon A\to B$ be a morphism of monoids. The following pair of conditions emerged naturally in my research: For all $a\in A$ and $b_1,b_2\in B$ such that $f(a)=b_1....
Gejza Jenča's user avatar
0 votes
1 answer
305 views

Embedding a semigroup into a divisible semigroup

The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
Salvo Tringali's user avatar
0 votes
0 answers
61 views

Defining rank of an abelian subgroup using the second centralizer

I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO. I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
dbossaller's user avatar
0 votes
0 answers
63 views

A construction that sort of merges two semigroups to build a new one

Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
Salvo Tringali's user avatar
0 votes
0 answers
250 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] ...
David Pokorny's user avatar