# New Identities In Universal Osborn Loops - Mathematics > General Mathematics

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Abstract: A question associated with the 2005 open problem of Michael Kinyon Is everyOsborn loop universal?, is answered. Two nice identities that characterizeuniversal left and right universal Osborn loops are established. Numerous newidentities are established for universal left and right universal Osbornloops like CC-loops, VD-loops and universal weak inverse property loops.Particularly, Moufang loops are discovered to obey the new identity$yx^{-1}u\cdot u^{-1}xu=yxu\cdot u^{-1}x^{-1}u$ surprisingly. Forthe first time, new loop properties that are weaker forms of well known loopproperties like inverse property, power associativity and diassociativity areintroduced and studied in universal left and right universal Osborn loops.Some of them are found to be necessary and sufficient conditions for auniversal Osborn to be 3 power associative. For instance, four of them arefound to be new necessary and sufficient conditions for a CC-loop to be powerassociative. A conjugacy closed loop is shown to be diassociative if and onlyif it is power associative and has a weak form of diassociativity.

Autor: Temitope Gbolahan Jaiyeola, John Olusola Adeniran

Fuente: https://arxiv.org/