>> There is commutativity for the identity element, but it is not needed to
>> be commutative.
>
> A monoid is /defined as/ having a two-sided identity.

I had hoped to prove that the associativity of # implies the 
two-sidedness of any identity element. Instead, somebody showed me a 
counter-example:

If we define # as

   ∀ x, y ∈ S, x # y = y

then every element of S is trivially a left-identity, and no 
right-identity exists. Miraculously, this operator is also associative:

   x # y # z
   (x # y) # z = y # z = z
   x # (y # z) = x # z = z

So it seems it /is/ perfectly possible for a semigroup to have one or 
more one-sided inverses - it's just that they all have to be /the same/ 
side. A monoid, on the other hand, is /defined as/ having a two-sided 
inverse, as I originally asserted.

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