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IMI/Publicaţii/QRS/Ediţii/QRS v.20, n.2 (28), 2012/

A new characterization of Osborn-Buchsteiner loops

Authors: T. G. Jaiyeola, J. O. Adeniran

Abstract

In the study of loops of Bol-Moufang types, a question that quickly comes to mind is this. Since a loop is an extra loop if and only if it is a Moufang loop and a CC-loop(or C-loop), then can one generalize this statement by identifying a "new identity" for a loop which generalizes the C-loop identity such that we can say "An Osborn loop is a Buchsteiner loop if and only if it obeys "certain" identity? A somewhat close answer to this question is the unpublished fact by M. K. Kinyon that "An Osborn loop Q with nucleus N is a Buchsteiner loop if and only if the quotient Q/N is a Boolean group" where Q/N being a Boolean group somewhat plays the role of the missing identity. It is proved that an Osborn loop is a Buchsteiner loop if and only if it satisfies some identity. The importance of its emergence which was traced from the facts that Buchsteiner loops generalize extra loops while Osborn loops generalize Moufang loops is the fact that not every Osborn-Buchsteiner loop is an extra loop. An LC-loop obeys this identity. Necessary and sufficient condition for a OBL to be central square is established.




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