  A question associated with the 2005 open problem of Michael Kinyon (Is every
Osborn loop universal?), is answered. Two nice identities that characterize
universal (left and right universal) Osborn loops are established. Numerous new
identities are established for universal (left and right universal) Osborn
loops like CC-loops, VD-loops and universal weak inverse property loops.
Particularly, Moufang loops are discovered to obey the new identity
$[y(x^{-1}u)\cdot u^{-1}](xu)=[y(xu)\cdot u^{-1}](x^{-1}u)$ surprisingly. For
the first time, new loop properties that are weaker forms of well known loop
properties like inverse property, power associativity and diassociativity are
introduced and studied in universal (left and right universal) Osborn loops.
Some of them are found to be necessary and sufficient conditions for a
universal Osborn to be 3 power associative. For instance, four of them are
found to be new necessary and sufficient conditions for a CC-loop to be power
associative. A conjugacy closed loop is shown to be diassociative if and only
if it is power associative and has a weak form of diassociativity.

General Mathematics;Group Theory