The Hausdorff series provides a solution to the equation w=\log(e^ue^v) given by a recursive formula which can be expressed as nested commutators of u and v.
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 L_m\times R_n determined by the left zero semigroup L_m and the right zero semigroup R_n of order m and n, respectively. Let \mathbb R\langle L_m\times R_n\rangle be the semigroup ring spanned on L_m\times R_n
together with the identity element 1. We provide a closed form of the formula for solving the equation in \mathbb R\langle L_m\times R_n\rangle.



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 Kelekci, Osman