A

__semigroup __is a non-empty set $\displaystyle S$ together with an associative binary operation $\displaystyle *$, $\displaystyle *:S \times S \rightarrow S$.

A

__rectangular band__ is a semigroup $\displaystyle S := A \times B$, $\displaystyle A, B$ non-empty sets, under the operation $\displaystyle (a_1, b_1)*(a_2, b_2) := (a_1, b_2)$ for all $\displaystyle a_i \in A, b_i \in B$.

1) Let $\displaystyle S$ be a semigroup such that $\displaystyle x^2 = x$ and $\displaystyle xyz = xz$ for all $\displaystyle x,y,z \in S$. Prove that $\displaystyle S$ is isomorphic to a rectangular band.