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The generators of total multiplication group of Cheban loop | ||
| Journal of Algebra and Related Topics | ||
| دوره 14، شماره 1، آبان 2026، صفحه 159-182 اصل مقاله (207.18 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2026.28508.1717 | ||
| نویسندگان | ||
| B. Osoba* 1؛ T. G. Jaiyeola2؛ L. O. Adekola1؛ M. T. Adenibuyan3 | ||
| 1Department of Physical Sciences, Bells University of Technology, Ota, Ogun State, Nigeria | ||
| 2Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria Department of Mathematics, University of Lagos, Akoka, Nigeria | ||
| 3Department of Computer Science, Bells University of Technology, Ota, Ogun State, Nigeria | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Cheban loop؛ Automorphic loop؛ Inner mapping group؛ Total multi[plication group | ||
| مراجع | ||
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