What's the best?

What's the best?

rings

magmas are p chill

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

Abelian groups because they are easy to work with

abelian groups because everything meaningful in mathematics forms abelian groups.

Why totality over closure? What's the difference?

Lmfao XD
Obviously categories

lol

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.

Groups of functions by composition retard.