Exercise
Suppose (R,+,*,0,1) is a structure consisting of a set R for which (R,+,0) is a commutative group and for which (R,*,1) is a commutative monoid. Does left associativity imply right associativity?
Yes.
No.
It depends.
Indeed,
proving
right distributivity.
| (y + z) * x | = (commutative *) |
| x * (y + z) | = (left distributivity) |
| x * y + x * z | = (commutative *) |
| y * x + z * x , |
Well, think again.
The answer is "yes" or "no".