Hausdorff series in semigroup rings of rectangular bands

"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 $\mathbb R\langle L_m\times R_n\rangle$."