Does associativity imply commutativity? I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea:
$$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ Here using associativity of addition operation, we've shown commutativity. 
In general, $$\underbrace{(1+1+\dots+1)}_{a \, 1\text{'s }}\ \ + \ \ \underbrace{(1+1+\dots+1)}_{b \, 1\text{'s }}=\underbrace{(1+1+\dots+1)}_{b \, 1\text{'s }}\ \ + \ \ \underbrace{(1+1+\dots+1)}_{a \, 1\text{'s }} \\ \implies a+b=b+a$$
For any natural $a,b$.
Hence using only associativity we prove commutativity. That this can be done, is disturbing me too much. Is this really correct? If yes, then are associativity and commutativity closely related? Or is it because of some other property of natural numbers? If yes, then can it be done for other structures as well?

Consider an $\operatorname L$ operator defined as 'take the left': $$a \operatorname L b = a$$
For any three $a,b,c$:
$$(a \operatorname L b) \operatorname L c = a \operatorname L c = a = a \operatorname L b = a \operatorname L (b \operatorname L c)$$
but
$$a \operatorname L b = a \ne b = b \operatorname L a$$
unless $a=b$.
