Search Results
Search type | Search syntax |
---|---|
Tags | [tag] |
Exact | "words here" |
Author |
user:1234 user:me (yours) |
Score |
score:3 (3+) score:0 (none) |
Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
Views | views:250 |
Code | code:"if (foo != bar)" |
Sections |
title:apples body:"apples oranges" |
URL | url:"*.example.com" |
Saves | in:saves |
Status |
closed:yes duplicate:no migrated:no wiki:no |
Types |
is:question is:answer |
Exclude |
-[tag] -apples |
For more details on advanced search visit our help page |
This tag is used if a reference is needed in a paper or textbook on a specific result.
5
votes
Dihedral extensions and the Ankeny–Artin–Chowla conjecture
It appears that my former officemate Andreas Reinhart (University of Graz, Austria) has disproven the Ankeny–Artin–Chowla conjecture: more precisely, Andreas has found that
$$
d := 331914313984493$$
i …
4
votes
1
answer
217
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, …
8
votes
1
answer
314
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 writ …
3
votes
Accepted
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative ...
Sorry for answering my own question, but YCor's construction in a related thread (here) gave me a lightbulb moment. Hopefully, it's not a broken lightbulb.
The answer to the question asked in the OP …
7
votes
2
answers
459
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative ...
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 ta …
8
votes
2
answers
541
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 …
3
votes
0
answers
78
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 $b = ab = ba$, …
9
votes
1
answer
408
views
On the origin of a fundamental theorem of additive number theory
Given $a, b \in \mathbb Z$, set $[\![a,b]\!] := \{x \in \mathbb Z: a \le x \le b\}$. A basic result in additive number theory goes as follows:
If $A$ is a finite subset of $\mathbb N$ with $0 \in A$ …
12
votes
Accepted
On the origin of a fundamental theorem of additive number theory
I shared the link to this thread with several colleagues, inviting them to contribute to the discussion. Notably, I received a response from Melvyn Nathanson himself, most of which is reproduced below …
8
votes
2
answers
560
views
Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?
A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on …
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 semig …
5
votes
0
answers
160
views
$S$ and $T$ globally isomorphic semigroups, with $S$ (commutative and) cancellative, iff $S$...
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 induced by $S$:
…
2
votes
0
answers
90
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).
…
2
votes
0
answers
213
views
Squares whose differences are squares
EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in Question 2 bel …
3
votes
1
answer
166
views
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative ra...
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 (m …