Basarab loop and the generators of its total multiplication group

A loop (Q; ·) is called a Basarab if the identities: (x · yxρ)(xz) = x yz and (yx) (xλz x) hold. It was shown that left, right middle nuclei of coincide, nucleus set elements whose inner mapping Tx are automorphisms. The generators group were refined in terms one total loop. Necessary su_cient condition(s) (associators) for to be established. discovered loop: ↦ an endomorphism only left (right) regular mapping. established automorphic mappings belong its group. (A-loop) it (middle A-loop). Some interesting relations involving multiplication derived, based on these, finetuned. totally (TA-loop) commutative exible These aforementioned results used give partial answer 2013 question ostensible solution 2015 problem case