What's the best?

rings

magmas are p chill

>>8938895
>What's the best?
to crush your enemies, see them driven before you, and to hear the lamentations of their women

>>8938895
Abelian groups because they are easy to work with

>>8938895
abelian groups because everything meaningful in mathematics forms abelian groups.

Why totality over closure? What's the difference?

>>8941189
Lmfao XD
>>8938895
Obviously categories

File: Screen Shot 2017-05-28 at 23.09.57.png (29KB, 916x375px) Image search: [Google]

29KB, 916x375px

>>8941189
lol

>>8941194
It's a bit unnecessary they even mention Closure here.

Consider first the abelian group Z3, which has a representation
({0,1,2}, +)
and
1+1 = 2
1+2 = 0
2+2 = 1
x+0 = 0 for all x
...
You can view this as a groupoid with a single object A with 3 arrows on it:
[math] id_0 : A \to A[/math]
[math] f_1 : A \to A[/math]
[math] f_2 : A \to A[/math]
and + being the arrow concatenation, with e.g. the rule
[math] f_2\, \circ\, f_2 = f_1 [/math]
representing
2+2=1
above


Now Consider instead following proper groupoid: It consists of two object A and B and two arrows going away from each.

[math]id_A : A \to A[/math]
[math]id_B : B \to B[/math]
[math] f_A : A\to B [/math]
[math] f_B : B\to A [/math]
with the rules
[math] f_A \circ f_B = id_B [/math]
[math] f_B \circ f_A = id_A [/math]

Here [math] \circ [/math] is now not total, because
[math] id_A \circ i_B [/math]
doesn't make sense.
It's also not a closed structure, because you move between object and it's e.g. not like [math] \circ [/math] maps two arrows of type [math] A\to B [/math] to another one of that type.

>>8941189
Groups of functions by composition retard.

[math] id_A \circ id_B [/math]

I meant
x+0 = x for all x

>>8941328
>Groups of functions by composition retard.
Literally useless

>>8941343
>permutations are useless

>>8941348
Lagrange was studying permutations way before the concept of group even existed, and if you know the work of Lagrange then you already know all of mathematics and physics so... pretty useless.

Give me one problem in the universe that can't be solved using Lagrange Multipliers.

GOT EM

Are people in this thread arguing that Lie groups like SO(3) are of no interest and don't simplify shit?

>logic
Nice mem tee bee aych.

Why are they called axioms when they're actually just definitions?

>>8941376
The aren't definitions here.

The logical sentence
[math] \forall a,b,c. \, (a*b)*c = a*(b*c) [/math]
can be taken to be one of many defining requirements of a theory (e.g. of abelian groups, or the additive structure of a ring) and there you call it axiom.

A definition would only be the name-giving of that rule viewed as a property of a structure. You may DEFINE a predicate "isAbelian" for groups by saying it's true if a group fulfills the above rule.

But you're right in that each axiom of a theory is one of it's "defining properties", if you will.
Axiom is just the name of kind of logical sentences taken to hold true, for characterizing a theory at hand.

>>8938895
Abelian has more green. Let's go with that one.

Is it possible to give a concise explanation of all structures in OP pic except groups and abelian groups (for those I already know)? I'm curious

>>8941432
If you know what abelian groups are, then the others shouldn't be that hard to figure out...

>>8941189
>What are matrices