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Right division in Moufang loops

Maria de Lourdes M. Giuliani; Kenneth Walter Johnson — 2010

Commentationes Mathematicae Universitatis Carolinae

If $(G, \cdot)$ is a group, and the operation $(*)$ is defined by $x * y = x \cdot y^{- 1}$ then by direct verification $(G, *)$ is a quasigroup which satisfies the identity $(x * y) * (z * y) = x * z$ . Conversely, if one starts with a quasigroup satisfying the latter identity the group $(G, \cdot)$ can be constructed, so that in effect $(G, \cdot)$ is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities produced by...

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