A Closed Formula of Hausdorff Series in a Semigroup Ring

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Let 𝐾 be a fileld of characteristic zero, 𝑢 and 𝑣 be algebraically independent variables and 𝐾〈𝑢,𝑣〉 be the free associative noncommutative algebra of rank 2 over 𝐾. Each element in 𝐾〈𝑢,𝑣〉 can be written as a noncommutative polynomial of 𝑢 and 𝑣. The expression of the polynomial 𝑤=𝑤(𝑢,𝑣)=log (𝑒𝑢𝑒𝑣) is a formal power series and a solution to this equation for 𝑤 is given by the Hausdorff series expressed as nested commutators of 𝑢 and 𝑣. However this series is not in its closed form in 𝐾〈𝑢,𝑣〉. Obtaining a closed form of this series, one may consider another algebraic structure other than 𝐾〈𝑢,𝑣〉 and evolute the series in it. We consider the right zero semigroup with finite elements, and the semigroup ring 𝐴 spanned on this semigroup over the field of real numbers. In this paper, we provide a closed form of this formula in the semigroup ring 𝐴.

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Hausdorff series , Right zero semigroup , Semigroup ring

Erciyes University Journal Of The Institute Of Science an‎d Technology

483

Erciyes University Journal Of The Institute Of Science an‎d Technology