Mathematics - 2019-09-22

Bob

01:19

Are there any Boolean algebra experts around?

Rithaniel

01:40

When trying to show that a group is free abelian, it's sufficient to show that it has a basis, right? Why is that enough?

Ted Shifrin

01:56

@Rithaniel: I haven't thought about this in a long time, but it depends on your definition of basis, I guess. You need to be sure there are no relations among your generators.

Rithaniel

Well, the definition we've been using is that a basis for a free abelian group is a set that generates the group, and that for any finite subset $\{x_i\}_{i\in I}$ of this generating set, $\sum\limits_{i\in I}n_ix_i=0$ with $n_i\in\mathbb{Z}$ implies that each $n_i=0$.

I should look up some proofs this weekend.

Ted Shifrin

02:34

Right. That's saying no relations.

Akiva Weinberger

03:45

@Rithaniel Heh, that's almost like "linearly independent" with vector spaces

except that, in $\Bbb Z_n$, every singleton (set with one element) is linearly dependent

because $nx=0$ but $x\ne0$

(Well, I suppose in a vector space, $\{0\}$ is linearly dependent.)

(So it's not so strange.)

(Just strange that non-identity elements could do that.)

Rithaniel

Yeah, basically, saying that a group has a nonempty basis is equivalent to saying that it's isomorphic to the direct sum of $x$ many copies of $\mathbb{Z}$

So, I suppose it might be the correct spirit to say that it's the same thing as a vector space, but instead of being over a field, it's over $\mathbb{Z}$.

Akiva Weinberger

Yeah

The word is "module"

A module is basically a vector space over a ring instead of a field

Rithaniel

Oh yeah, I've heard of those. It's a vector space, but for-- ninja'd.

Akiva Weinberger

Yeah so an abelian group is the same concept as a module over $\Bbb Z$

I don't know how bases work for modules in general though

All I know is the word

Rithaniel

04:00

Well, the definition of a basis we've been using seems to force the object the basis if for to specifically be a free abelian group.

Akiva Weinberger

I've heard the phrase "Grobner basis"

It might be related, I dunno

I don't actually know what that is

but it has 'basis' in it so it might be related

Rithaniel

Right now, my brain is on the topic of infinitely generated abelian groups that are not free abelian groups. I want to say that $\mathbb{Q}$ under addition is an example.

Akiva Weinberger

Makes sense

Everything can be divided by two, so whatever you have in your basis can be written as the sum of smaller things

Rithaniel

Apparently, if you have that $G$ is abelian, finitely generated and, each element has infinite order, then $G$ is free abelian. However, if $G$ is abelian, not finitely generated, and each element has infinite order, then it isn't free abelian. I'm trying to exactly hone in on what is changing between the two.

(I mean, obviously what is changing is that you need infinitely more generators for one than the other, but I mean in a more philosophical sense)

Akiva Weinberger

If @Secret were here he'd ask what if the generating set were Dedekind-finite

Rithaniel

04:09

Anyways, what've you been up to, lately, Akiva?

Akiva Weinberger

A bunch of Japanese assignments for next week were posted online

so I'm trying to get them all in at once

using the textbook

Rithaniel

That's how you do it. This HW I'm working on now isn't due until Friday.

Akiva Weinberger

There's a test on Monday

Rithaniel

Ah, in Japanese?

Akiva Weinberger

@Rithaniel Yeah

Gotta know those charts^ plus some vocab and grammar

Every syllable has two characters

Rithaniel

04:12

I wish you luck on it. I'm terrible when trying to learn other languages.

Akiva Weinberger

and there's not much similarity between the two syllabaries

Rithaniel

Well, Shi and Tsu are currently my favorites.

Akiva Weinberger

Maybe better picture ('cause they're side-by-side)

@Rithaniel シ and ツ?

Rithaniel

Yes, the faces

Akiva Weinberger

ソ and ン (so and n) are great too

because they're essentially identical

and make you struggle for half a minute trying to read パソコン

(It's pasokon - short for 'perso'nal 'com'puter)

(ハ ha plus circle equals パ pa)

Rithaniel

04:17

I wonder if people learning English have the same sort of issues with P, B, and R.

Akiva Weinberger

b d p and q are the same letter

If <q> were pronounced /t/ it would be so much better

because then we'd be able to say "stem up = voiced, stem down = voiceless"

But no <q> ruins the pattern

Rithaniel

Just rotated $\pi$ radians or reflected about the horizontal axis.

Akiva Weinberger

Rotating 180 degrees is a point reflection about the origin

b <-> q

p <-> d

not a reflection across an axis

Rithaniel

Ah right, should be horizontal reflection

b <-> p
q <-> d

Akiva Weinberger

Fun fact: the Japanese word for bread is 'pan' - a Portuguese loanword

because there is no wheat natively in East Asia

(pretty sure anyway)

The staple crop is rice

(The word for "meal" also is the same as the word for "white rice", incidentally)

(Kinda like lechem in Biblical Hebrew meaning both "bread" and "food")

Nice when little cultural things come through in the language like that

Rithaniel

04:23

Yeah, it's interesting how culture affects language

Akiva Weinberger

And both are malleable

Culture changes over time

and with it, language

Rithaniel

I wonder how much the more strange traits of languages come from culture. Like phonetic inconsistency, as an example.

Akiva Weinberger

Phonetic inconsistency?

Rithaniel

Like through and bough

Akiva Weinberger

Oh

I would call that orthographic inconsistency

I guess it depends on your perspective

Recently saw this thing: zompist.com/yingzi/yingzi.htm

"If English was written like Chinese"

Rithaniel

04:28

Well that's an interesting thought experiment

Akiva Weinberger

> the volve in revolve, evolve, involve, devolve, which would have its own yingzi, and would seem as much a "word" or component of the language as the match in rematch, mismatch, unmatch

> Worse yet, the -cuit of biscuit and circuit might be written with the same character (a derivative of kit), and a meaning sought for it-- perhaps 'round', since biscuits are round and circuits involve going round. Again, etymologically this is nonsense.

> There would even be a tendency to think of words as derived from characters rather than the other way around.

Érico Melo Silva

@AkivaWeinberger much more interestingly is the fact that tempura is also a loanword

Akiva Weinberger

Is it really?

Rithaniel

Perhaps the words ending in "-morphism" would end in "-volve"

Akiva Weinberger

> From Portuguese, ultimately from Latin. Different dictionaries link two different original terms:

> Portuguese tempero (“seasoning”) or tempera (“he/she/it seasons; season!”), third-person present singular or imperative tense of temperar (“to season, to temper”), from Latin temperare (“to mix, to temper”).[1][2][3]
> Portuguese têmpora (“Ember days”), from Latin tempora, plural of tempus (“time; period”). When Portuguese explorers (mostly Jesuit missionaries) arrived in Japan, they abstained from eating beef, pork and poultry during the Ember days series of holidays. Instead, they ate fr…

Érico Melo Silva

04:30

temperar means to season but also sometimes to cook

Akiva Weinberger

Interesting

Ted Shifrin

Hi, @Eric, DogAteMy, re @Rithaniel.

Érico Melo Silva

hi

Rithaniel

Heya Ted

Ted Shifrin

@Rithaniel: Relative to your earlier comment, you might want to investigate the difference between direct sum and direct product ...

Rithaniel

04:45

Ooooo, yeah. So if you have an abelian, infinitely generated group such that each element has infinite order, it might be the direct product of infinitely many copies of $\mathbb{Z}$?

Semiclassical

05:10

I get way too interested sometimes in where names of math things come from

case in point, my comments here: math.stackexchange.com/questions/3363419/…

Akiva Weinberger

07:16

Tautology club meets at least twice a week, if not less

Leaky Nun

@AkivaWeinberger well if it isn't less than twice a week then it must be at least twice a week!

Sayan Chattopadhyay