I'm working on a question on complex function with binary operation, it has two parts in the question and I got stuck on part b.
Part a: represent the $∘$ operation in a graphical way.
I have drawn a graph for $z_a∘z_b=R(z_a)+iI(z_b)$ from my understanding. ( am I right?)
I am told that my graph is wrong, below is my new graph.
Part b: consider the set $C$ together with binary operation $∘$ defined by:
$z_a∘z_b=f(z_a,z_b)$, where $f(z_a,z_b)=R(z_a)+iI(z_b)$.
The set $C$ together with the binary operation $∘$ forms a ? ( choose one from the followings)
a) Monoid b) Semigroup c) Group d) Magma e) Ring f) Field
$R(z_a)$ is the real part of $z_a$ and $I(z_b)$ is the imaginary part of $z_b$ I know it satisfies closure and associativity such that $f(z_a,f(z_b,z_c))=f(f(z_a,z_b),z_c)$ so it is at least a semigroup.
I'm very uncertain for my answer because by looking on the graph, I think it has identity and inverse elements which suggests it can be field. I don't think it is a ring because it is clearly not an Abelian group ( a$∘$b is not same as b$∘$a).
Any help will be appreciated. Thanks.