I am reading some notes in which I found the following exercise:
Suppose $G$ is a magma then $G$ is associative and satisfy cancellation properties.
I think this is not true for instance matrix multiplication does not satisfy the cancellation property so I think some assumption is missing from the exercise. What could be possible minimum assumption to make the question meaningful.
Classically, a magma is just a set with a binary operation. This operation has no reason, without more hypothesis, to be associative or to satisfy cancellation properties.