Le 23/04/2012 12:19, Invisible a écrit :
> A magma isn't especially interesting by itself. However, if we add the
> rule that # must be /associative/, then we got a "semigroup".
> 
>   ∀ x, y, z ∈ S, (x # y) # z = x # (y # z).
> 
> Already this begins to have interesting mathematical properties. But if
> we add a special /identity element/, we get a "monoid".
> 
>   ∃ i ∈ S: ∀ x ∈ S, i # x = x # i = x.
> 

Hey, not only you added identity element, but you seems to imply also
(but not really) that its also commutative:

∀ x ∈ S, ∀ y ∈ S, x # y = y # x.

So please be more careful for my poor brain and write instead:

∃ i ∈ S: ∀ x ∈ S, i # x = x and x # i = x.

There is commutativity for the identity element, but it is not needed to
be commutative.

> If you now take your monoid and give every element a corresponding
> /inverse element/, then you have a "group".
> 
>   ∀ x ∈ S, ∃ y ∈ S: x # y = y # x = i.

Your definition of inverse element once again seems to imply general
commutativity, it is not needed.

(I like to think about the Quaternion group (non abelian group of order
8): 1 & -1 are their own inverse, but inverse of i,j,k are same negative
(-i, -j, -k))

Yes, -i.i = i.-i = 1
but i.j = -j.i = k

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