Hausdorff series in semigroup rings of rectangular bands

Kelekci, Osman

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creative_2023_32_1_49_54

DOI: https://doi.org/10.37193/CMI.2023.01.06

Issue no: Vol 32/2023 no. 1 Tags: Hausdorff series, rectangular band, semigroup algebra

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