Variations of some inverse properties in Cheban loops [Articol]

Abstract

A Cheban loop (G; ) is characterized by the identities (z yx)x = zx xy and x(xy z) = yx xz. It was shown that a Cheban loop (G; ) satisfies the cross inverse property if and only if (G; ) is an abelian group or all the left (right) translations of (G; ) are right (left) regular. Additionally, it was proved that in a Cheban loop, the following algebraic properties are equivalent: flexibility, left inverse property, right inverse property, inverse property, left alternativity, right alternativity and alternativity. This study also revealed that a Cheban loop could be categorized as a weak inverse property loop if it is flexible and the middle inner mapping belongs to a permutation group; as an automorphic inverse property loop if it has a semi-commutative law and the middle inner mapping is contained in a permutation group; as an anti-automorphic inverse property loop if every element has a two-sided inverse and the middle inner mapping is contained in a permutation group; and as a semi-automorphic inverse property loop if it is flexible, the middle inner mapping is contained in a permutation group, and a semi-cross inverse property holds. Finally, the study established the necessary and sufficient conditions for a Cheban loop to have an exponent of 2 or to be a centrum square.