fix and define the map by clearly is well-defined.

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:

and therefore