Is this a*b = a is semigroup and commutative?

Question

Let S be any nonempty set with the operation a * b = a. Is (S,*) a semigroup? Is it commutative?

I dont know what to do if a * b = merely a. Usually if a * b = anything that have operation i know how to do. Can anyone show the proof as well? Thanks.

Answer 1

$(S,*)$ is a semigroup but the operation is not commutative. To be a semigroup the operation $*$ must be associative, which it is. Let $a,b,c \in S$ then consider

$$a*(b*c) = a* b = a$$ and $$ (a*b)*c = a*c = a$$ thus $a*(b*c)\equiv(a*b)*c$. It is not commutative however, consider two elements $a,b\in S:a\neq b$. $$a*b = a \neq b = b*a.$$