1) preserves multiplication: this is equivalent to for all which is given in the problem.
2) is injective: suppose i.e. then and thus
3) is surjective: suppose then
the non-trivial side: suppose we claim that and for all and therefore we're done by the first part of the problem.
2) Prove that a semigroup is a rectangular band if and only if for all we have that .
1) : by associativity we have and thus
2) : let since we have and so for all thus hence: