The generators of total multiplication group of Cheban loop

Osoba, B.; Jaiyeola, T. G.; Adekola, L. O.; Adenibuyan, M. T.

doi:10.22124/jart.2026.28508.1717

The generators of total multiplication group of Cheban loop

Articles in Press

Document Type : Research Paper

Authors

¹ Department of Physical Sciences‎, ‎Bells University of Technology‎, ‎Ota‎, ‎Ogun State‎, ‎Nigeria

² Department of Mathematics‎, ‎Obafemi Awolowo University‎, ‎Ile Ife 220005‎, ‎Nigeria Department of Mathematics‎, ‎University of Lagos‎, ‎Akoka‎, ‎Nigeria

³ Department of Computer Science‎, ‎Bells University of Technology‎, ‎Ota‎, ‎Ogun State‎, ‎Nigeria

10.22124/jart.2026.28508.1717

Abstract

A Cheban loop $(G‎, ‎\circ)$ is characterized by the identities $(z\circ yx)x= zx\circ xy$ and $x(xy\circ z)= yx\circ xz$ for all $x,y,z\in G$‎. ‎It was‎ ‎established that the left‎, ‎right‎, ‎and middle nuclei of a Cheban loop coincide‎, ‎and the nucleus of a Cheban loop is the set of elements $a$ whose middle inner mappings $T_a$ are automorphisms‎. ‎The generators of the inner mapping group of a Cheban were refined in terms of one of the generators of the total inner mapping group of a Cheban loop‎. ‎Necessary and sufficient conditions regarding the inner mapping group (associators) for a loop to be a Cheban loop were established‎. ‎It was shown that‎, ‎in a Cheban loop‎, ‎the mapping $a\mapsto T_a$ is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping‎. ‎Additionally‎, ‎a Cheban loop was proved to be a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group‎. ‎Furthermore‎, ‎a Cheban loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop)‎. ‎Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Cheban loop were derived‎, ‎and based on these‎, ‎the generators of the total inner mapping group of a Cheban loop were fine-tuned‎. ‎Finally‎, ‎it was shown that a Cheban loop is a totally automorphic loop (TA-loop) if and only if it is a commutative and flexible loop‎. ‎These results above were used to give a partial answer to a 2013 question and an apparent solution to the 2015 problem in the case of a Cheban loop‎.

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