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"The Hausdorff series provides a solution to the equation w=log(eᵘeᵛ) 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 Lₘ× Rₙ determined by the left zero semigroup Lₘ and the right zero semigroup Rₙ of order m and n, respectively. Let R Lₘ× Rₙ be the semigroup ring spanned on Lₘ× Rₙ together with the identity element $1$. We provide a closed form of the formula for solving the equation in R Lₘ× Rₙ."
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OSMAN KELEKCI (2022) studied this question.
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