Characterization of some subloops of a Basarab loop

Abstract

A loop (Q, ·) is called a Basarab loop if the identities: (x · yx^ρ) · (xz) = x· yz and (yx) · (x^λz · x) = yz · x hold. A Basarab loop was shown to be a group if and only if it is a cross inverse property loop or its middle inner mapping is automorphic in action.
Some algebraic properties of the left, right and middle inner mappings of a Basarab loop were explored. The algebraic properties of some bi-variate functions defined on a Basarab loop Q were established and these were used to explore the properties of
some important mono-variate functions defined on Q. In addition, these bi-variate functions aided the establishment of fine-tuned necessary and sufficient conditions for
a loop to be a left (right) Basarab loop, and Basarab loop in relations to left (right) CC-loop, and CC-loop, respectively. Some subloops of a Basarab loop were shown to
be characterized by the mono-variate functions. New characterizations of the centrum of a Basarab loop were obtained.