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8 votes
1 answer
437 views

Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$

I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
gmvh's user avatar
  • 3,065
11 votes
0 answers
427 views

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 ...
Keith's user avatar
  • 601
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
4 votes
1 answer
239 views

True or false? Every left or right cancellative, duo semigroup is cancellative

A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
Salvo Tringali's user avatar
8 votes
1 answer
322 views

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 ...
Salvo Tringali's user avatar
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,...
Salvo Tringali's user avatar
7 votes
2 answers
488 views

Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?

By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
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
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
3 votes
1 answer
171 views

Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational

Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
Salvo Tringali's user avatar
6 votes
1 answer
173 views

References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994). I would like to ask for references on semigroup actions on ...
Marco Farotti's user avatar
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
5 votes
0 answers
160 views

$S$ and $T$ globally isomorphic semigroups, with $S$ (commutative and) cancellative, iff $S$ is isomorphic to $T$?

Denote by $\mathcal P(S)$ the semigroup obtained by equipping the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication ...
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
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
4 votes
1 answer
151 views

Three preprints and one manuscript of Tamura on power semigroups

I'm reading Takayuki Tamura's article "On the recent results in the study of power semigroups", pp. 191-200 in Goberstein & Higgins' Semigroups and Their Applications, Kluwer, 1987 (the ...
Salvo Tringali's user avatar
6 votes
0 answers
259 views

Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki [1] defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
Duchamp Gérard H. E.'s user avatar
6 votes
1 answer
236 views

Two numerical monoids are isomorphic iff they are equal

A numerical monoid (or numerical semigroup) is a submonoid $S$ of the additive monoid $(\mathbb N, +)$ of non-negative integers with the property that the set $\mathbb N \setminus S$ is finite. It is ...
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
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
6 votes
3 answers
393 views

Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$)

Question. Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons? A monoid with ...
Salvo Tringali's user avatar
4 votes
0 answers
157 views

On skew monoid rings and skew ordered series rings

To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
Salvo Tringali's user avatar
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
6 votes
3 answers
411 views

Problem 0.9.10 in Cohn's "Free Ideal Rings and Localization in General Rings" (CUP, 2006)

Let $S$ be a monoid. On p. xvii of P.M. Cohn's Free Ideal Rings and Localization in General Rings (CUP, 2006), one reads that an element $u \in S$ is regular if (quote) "[...] it can be ...
Salvo Tringali'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
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
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
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
3 votes
1 answer
263 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_{\...
xir's user avatar
  • 2,044
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
1 answer
173 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 ...
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
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
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
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
4 votes
1 answer
169 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
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
5 votes
1 answer
597 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
3 votes
1 answer
226 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 ...
Jeff Strom's user avatar
  • 12.5k
6 votes
0 answers
89 views

Maximal number of commuting functions of a finite set

Let $S$ be a finite set with $n$ elements and let $F_S$ denote the set of functions from $S$ to $S$. I wonder whether anything is known about the maximal cardinality of a commuting subset of $F_S$? A ...
Anthony Quas's user avatar
  • 23.2k
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
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
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
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
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
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