Asemigroupis a non-empty set together with an associative binary operation , .

Arectangular bandis a semigroup , non-empty sets, under the operation for all .

1) Let be a semigroup such that and for all . Prove that is isomorphic to a rectangular band.

Spoiler:

2) Prove that a semigroup is a rectangular band if and only if for all we have that .