Basarab loop and the generators of its total multiplication group

Abstract A loop (Q, ·) is called a Basarab loop if the identities: (x·yxρ)(xz) = x· yz and (yx)·(xλz ·x) = yz ·x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner map...

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Autores principales:	Jaiyéọlá,T. G., Effiong,G. O.
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2021
Materias:	Basarab loops Inner mapping group Automorphic loop (A-loop)
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172021000400939
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spelling	oai:scielo:S0716-091720210004009392021-08-12Basarab loop and the generators of its total multiplication groupJaiyéọlá,T. G.Effiong,G. O. Basarab loops Inner mapping group Automorphic loop (A-loop) Abstract A loop (Q, ·) is called a Basarab loop if the identities: (x·yxρ)(xz) = x· yz and (yx)·(xλz ·x) = yz ·x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and sufficient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle Aloop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and flexible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.40 n.4 20212021-01-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172021000400939en10.22199/issn.0717-6279-4430
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Basarab loops Inner mapping group Automorphic loop (A-loop)
spellingShingle	Basarab loops Inner mapping group Automorphic loop (A-loop) Jaiyéọlá,T. G. Effiong,G. O. Basarab loop and the generators of its total multiplication group
description	Abstract A loop (Q, ·) is called a Basarab loop if the identities: (x·yxρ)(xz) = x· yz and (yx)·(xλz ·x) = yz ·x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and sufficient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle Aloop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and flexible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.
author	Jaiyéọlá,T. G. Effiong,G. O.
author_facet	Jaiyéọlá,T. G. Effiong,G. O.
author_sort	Jaiyéọlá,T. G.
title	Basarab loop and the generators of its total multiplication group
title_short	Basarab loop and the generators of its total multiplication group
title_full	Basarab loop and the generators of its total multiplication group
title_fullStr	Basarab loop and the generators of its total multiplication group
title_full_unstemmed	Basarab loop and the generators of its total multiplication group
title_sort	basarab loop and the generators of its total multiplication group
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2021
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172021000400939
work_keys_str_mv	AT jaiye7885latg basarabloopandthegeneratorsofitstotalmultiplicationgroup AT effionggo basarabloopandthegeneratorsofitstotalmultiplicationgroup
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Basarab loop and the generators of its total multiplication group

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