# Holomorphy of Basarab Loops

## Main Article Content

## Abstract

A loop (*Q*,∙) is called a Basarab loop if the identities: (*x* ∙ *y**x ^{ρ}*)∙(

*xz*)=

*x*∙

*yz*and (

*yx*) ∙ (

*x*

^{λ}*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*).

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## Article Details

*Gulf Journal of Mathematics*,

*17*(1), 142-166. https://doi.org/10.56947/gjom.v17i1.2076