I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the others and vice versa? Thanks for your time and your consideration.
If you have a function $f_1 : S_1 \rightarrow S_2$ and can show that $f_1(u) = f_1(v)$ for all $(u, v) \in R$ then you will have shown that $f_1$ is a homomorphism. If you can find such a function $f_1$ and another function $f_2 : S_2 \rightarrow S_1$ such that $f_2(u) = f_2(v)$ for all $(u, v) \in T$ and additionally show that they are either (i) both injective or (ii) both surjective then you will have proved isomorphism.