Connections between Basarab and Buchsteiner Loops

Basarab loops and Buchsteiner loops are both G-loops with deep algebraic andstructural properties. Extra loops belong to these two classes. This paper examinesthe main connections between these two classes of loops and examines the relationshipbetween Basarab loops and some other important classes of loops like: Osborn loop,conjugacy closed loop (CC-loop), Wilson loop, Moufang loop and extra loop. It wasestablished that Basarab loop is a class of loops which answers a cogent questionraised by Goodaire and Robinson in 1990: ‘Are there other equationally defined (andnaturally characterized) classes of G-loops (like the classes Wilson loops and extraloops) which are contained in the class of all conjugacy closed loops?’. It was shownthat the center and centrum of a Basarab loop coincide. A Basarab loop was foundto be both left A-loop and a right A-loop, and shown to be an inverse property loopif and only if its left and right inner mappings are commutators of left and righttranslations. The left and right inner mappings of a Basarab loop were proved to benuclear automorphisms. The following properties were discovered to be equivalent toeach other in a Basarab loop: flexibility, diassociativity, right inverse property, leftinverse property, inverse property, right alternative property, left alternative property,alternative property, anti-automorphic inverse property, any Moufang law and anyextra law.