Holomorphy of Basarab Loops

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Gideon Okon Effiong
Temitope Gbolahan Jaiyeola
Martin Chucks Obi
Lukman Shina Akinola

Abstract

A loop (Q,∙) is called a Basarab loop if the identities: (xyxρ)∙(xz)=xyz and (yx) ∙ (xλ zx)=yzx 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 αA(Q) with four autotopic characterizations actually form normal subgroups of A(Q).

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How to Cite
Effiong, G. O., Jaiyeola, T. G., Obi, M. C., & Akinola, L. S. (2024). Holomorphy of Basarab Loops. Gulf Journal of Mathematics, 17(1), 142-166. https://doi.org/10.56947/gjom.v17i1.2076
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