<abstract xmlns="http://www.w3.org/1999/xhtml">Let (<italic>L</italic>, ·) be any loop and let <italic>A</italic>(<italic>L</italic>) be a group of automorphisms of (<italic>L</italic>, ·) such that <italic>α</italic> and <italic>φ</italic> are elements of <italic>A</italic>(<italic>L</italic>). It is shown that, for all <italic>x, y, z</italic> ∈ <italic>L</italic>, the <italic>A</italic>(<italic>L</italic>)-holomorph (<italic>H, ○</italic>) = <italic>H</italic>(<italic>L</italic>) of (<italic>L</italic>, ·) is an Osborn loop if and only if <italic>xα</italic>(<italic>yz · xφ</italic>−1) = <italic>xα</italic>(<italic>yxλ · x</italic>) · <italic>zxφ</italic>−1. Furthermore, it is shown that for all <italic>x</italic> ∈ <italic>L</italic>, <italic>H</italic>(<italic>L</italic>) is an Osborn loop if and only if (<italic>L</italic>, ·) is an Osborn loop, (<italic>xα· xρ</italic>)<italic>x</italic> = <italic>xα</italic>, <italic>x</italic>(<italic>xλ · xφ</italic>−1) = <italic>xφ</italic>−1 and every pair of automorphisms in <italic>A</italic>(<italic>L</italic>) is nuclear (i.e. <italic>xα·xρ, xλ ·xφ</italic> ∈ <italic>N</italic>(<italic>L</italic>, ·)). It is shown that if <italic>H</italic>(<italic>L</italic>) is an Osborn loop, then <italic>A</italic>(<italic>L</italic>, ·) = 𝒫(<italic>L</italic>, ·)<italic>∩</italic>Λ(<italic>L</italic>, ·)<italic>∩</italic>Φ(<italic>L</italic>, ·)<italic>∩</italic> Ψ(<italic>L</italic>, ·) and for any <italic>α</italic> ∈ <italic>A</italic>(<italic>L</italic>), 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/awutm-2015-0016_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>α</mi><mo>=</mo><msub><mi>L</mi><mrow><mi>e</mi><mi>π</mi></mrow></msub><mo>=</mo><msubsup><mi>R</mi><mrow><mi>e</mi><mi>ϱ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></math><tex-math>$\alpha = L_{e\pi } = R_{e\varrho }^{ - 1}$</tex-math></alternatives></inline-formula>
 for some <italic>π</italic> ∈ Φ(<italic>L</italic>, ·) and some <italic>ϱ</italic> ∈ Ψ(<italic>L</italic>, ·). Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is <italic>A</italic>(<italic>L</italic>)) and the nucleus of (<italic>L</italic>, ·).</abstract>

Let (L, ·) be any loop and let A(L) be a group of automorphisms of (L, ·) such that α and φ are elements of A(L). It is shown that, for all x, y, z ∈ L, the A(L)-holomorph (H, ○) = H(L) of (L, ·) is an Osborn loop if and only if xα(yz · xφ−1) = xα(yxλ · x) · zxφ−1. Furthermore, it is shown that for all x ∈ L, H(L) is an Osborn loop if and only if (L, ·) is an Osborn loop, (xα· xρ)x = xα, x(xλ · xφ−1) = xφ−1 and every pair of automorphisms in A(L) is nuclear (i.e. xα·xρ, xλ ·xφ ∈ N(L, ·)). It is shown that if H(L) is an Osborn loop, then A(L, ·) = 𝒫(L, ·)∩Λ(L, ·)∩Φ(L, ·)∩ Ψ(L, ·) and for any α ∈ A(L), 
α=Leπ=Reϱ−1$\alpha = L_{e\pi } = R_{e\varrho }^{ - 1}$
 for some π ∈ Φ(L, ·) and some ϱ ∈ Ψ(L, ·). Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is A(L)) and the nucleus of (L, ·).

Holomorphy of Osborn loops

Department of Mathematics, Ambrose Alli University, Ekpoma, 310001, Nigeria

Department of Mathematics, Federal University of Agriculture, Abeokuta 110101, Nigeria and UNESCO Professorial Chair of Mathematics (Algebra), National Mathematical Centre, Abuja, Nigeria.

Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria; Department of Mathematics and Statistics, Bowen University, Iwo, Nigeria

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Holomorphy of Osborn loops"}

Let (L, ·) be any loop and let A(L) be a group of automorphisms of (L, ·) such that α and φ are elements of A(L). It is shown that, for all x, y, z ∈ L, the...

Holomorphy of Osborn loops

Published Online: Apr 09, 2016

Page range: 81 - 98

Received: Jan 02, 2015

Accepted: Jan 14, 2016

DOI: https://doi.org/10.1515/awutm-2015-0016

Keywords
Osborn loops, holomorphy

© 2015 Annals of West University of Timisoara - Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Holomorphy of Osborn loops

Published Online: Apr 09, 2016

Page range: 81 - 98

Received: Jan 02, 2015

Accepted: Jan 14, 2016

DOI: https://doi.org/10.1515/awutm-2015-0016

KeywordsOsborn loops, holomorphy

© 2015 Annals of West University of Timisoara - Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Osborn loops, holomorphy