The Hausdorff series provides a solution to the equation given by a recursive formula which can be expressed as nested commutators of and .
Evolutions of the Haussdorff series in various algebras and rings has been considered in obtaining a closed form of this formula. We consider the rectangular band determined by the left zero semigroup and the right zero semigroup of order and , respectively. Let be the semigroup ring spanned on
together with the identity element . We provide a closed form of the formula for solving the equation in .