The generators of total multiplication group of Cheban loop

A Cheban loop (G‎, ‎\circ) is characterized by the identities (z\circ yx)x= zx\circ xy and x(xy\circ z)= yx\circ xz for all x,y,z\in G‎. ‎It was‎ ‎established that the left‎, ‎right‎, ‎and middle nuclei of a Cheban loop coincide‎, ‎and the nucleus of a Cheban loop is the set of elements a whose middle inner mappings T_a are automorphisms‎. ‎The generators of the inner mapping group of a Cheban were refined in terms of one of the generators of the total inner mapping group of a Cheban loop‎. ‎Necessary and sufficient conditions regarding the inner mapping group (associators) for a loop to be a Cheban loop were established‎. ‎It was shown that‎, ‎in a Cheban loop‎, ‎the mapping a\mapsto T_a is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping‎. ‎Additionally‎, ‎a Cheban loop was proved to be a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group‎. ‎Furthermore‎, ‎a Cheban loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop)‎. ‎Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Cheban loop were derived‎, ‎and based on these‎, ‎the generators of the total inner mapping group of a Cheban loop were fine-tuned‎. ‎Finally‎, ‎it was shown that a Cheban loop is a totally automorphic loop (TA-loop) if and only if it is a commutative and flexible loop‎. ‎These results above were used to give a partial answer to a ۲۰۱۳ question and an apparent solution to the ۲۰۱۵ problem in the case of a Cheban loop‎.

کلیدواژه ها:

Cheban loop ، Automorphic loop ، Inner mapping group ، Total multi[plication group