Holomorphy of Basarab Loops

Abstract

A loop (Q,∙) is called a Basarab loop if the identities: (x ∙ yxρ)∙(xz)=x ∙ yz and (yx) ∙ (xλ z∙x)=yz ∙ x hold. The holomorphy of a Basarab loop Q was investigated with respect to a group A(Q)$ of automorphisms of the loop. Some necessary and sufficient conditions for an A(Q)-holomorph of a loop Q to be a left (right) Basarab loop or Basarab loop were established. Specifically, the A(Q)-holomorph of a loop Q was shown to be a left (right) Basarab loop if and only if Q is a left (right) Basarab loop and every element of A(Q) is left (right) regular. The A(Q)-holomorph of a loop Q was shown to be a Basarab loop if and only if Q is a Basarab loop, every element of A(Q) is both left and right nuclear and the A(Q)-generalized inner mappings of Q take some particular forms. These results were expressed in form of commutative diagrams. In any left (right) Basarab loop or Basarab loop Q, it was shown that the set of α ∈ A(Q) with four autotopic characterizations actually form normal subgroups of A(Q).