The generators of total multiplication group of Cheban loop

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

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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References

[1] A. M. Cheban, Loops with identity of length of four and of rank three II, General Algebra and Discrete Geometry, 13 (1980), 117–120.
[2] A. I. Chinaka, A. O. Abdulkareem and J. O. Adeniran, Construction of right Cheban loops of small order, Generalised Lie Theory and Applications, 16 (2022), 1–6.
[3] A. I. Chinaka, A. O. Abdulkareem and J. O. Adeniran, Holomorph of right Cheban loops, ROMAI Journal, 19 (2023), 1–12.
[4] A. I. Chinaka, A. O. Abdulkareem, B. Osoba, Y. T. Oyebo and J. O. Adeniran, Bryant–Schneider group of right Cheban loop, Kragujevac Journal of Mathematics, (9) 50 (2026), 1413–1424.
[5] B. Cote, B. Harvill, M. Huhn and A. Kirchman, Classification of loops of generalised Bol–Moufang type, Quasigroups and Related Systems, 19 (2010), 193–206.
[6] T. G. Jaíyéọlá, A study of new concepts in smarandache quasigroups and loops, ProQuest Information and Learning(ILQ), Ann Arbor, USA, 2009.
[7] T. G. Jaiyéọlá, Smarandache isotopy of second Smarandache Bol loops, Scientia Magna Journal, (1) 7 (2011), 82–93.
[8] T. G. Jaiyéọlá, On Smarandache Bryant–Schneider group of a Smarandache loop, International Journal of Mathematical Combinatorics, 2 (2008), 51–63.
[9] T. G. Jaiyéọlá and G. O. Effiong, Basarab loop and the generators of its total multiplication group, Proyecciones, (4) 40 (2021), 939–962.
[10] T. G. Jaiyéọlá and O. O. George, On loops that satisfy x  (x  yx)z = (x  xy)  xz, Commentationes Mathematicae Universitatis Carolinae, Accepted.
[11] T. G. Jaiyeola, O. O. George, O. Ajayi and A. J. Saka, Mate I loop, Quasigroup and Related Systems, Accepted.
[12] T. G. Jaiyéọlá, O. O. George, B. Osoba and E. Ilojide, A class of power associative LCCloops and some associated total inner mapping group questions, Algebras, Groups and Geometries, (1) 41 (2025), 45–65.
[13] M. K. Kinyon, Totally automorphic loops, The Third Mile High Conference on Nonassociative Mathematics, University of Denver, Denver, Colorado, USA, August 11–17 (2013).
[14] B. Osoba and Y. T. Oyebo, On relationship of multiplication groups and isostrophic quasigroups, International Journal of Mathematics Trends and Technology (IJMTT), (2) 58 (2018), 80–84.
[15] B. Osoba, Smarandache nuclei of second Smarandache Bol loops, Scientia Magna Journal, (1) 17 (2022), 11–21.
[16] B. Osoba and T. G. Jaiyéọlá, Algebraic connections between right and middle Bol loops and their cores, Quasigroups and Related Systems, 30 (2022), 149–160.
[17] B. Osoba and Y. T. Oyebo, On the core of second Smarandache Bol loops, International Journal of Mathematical Combinatorics, 2 (2023), 18–30.
[18] B. Osoba, T. G. Jaiyéọlá and A. O. Abdulkareem, Cheban loop and its variance with some inverse properties, Quasigroups and Related Systems, 33 (2025), 95–106.
[19] T. Y. Oyebo, B. Osoba and T. G. Jaiyéọlá, Crypto-automorphism group of some quasigroups, Discussiones Mathematicae–General Algebra and Applications, 44 (2024), 57–72.
[20] H. O. Pflugfelder, Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann Verlag, Berlin, viii+147 pp., 1990.
[21] J. D. Phillips and V. A. Shcherbacov, Cheban loops, Journal of Generalised Lie Theory, (2011), 1–6.
[22] V. A. Shcherbacov, Elements of Quasigroup Theory and Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2017.
[23] D. Stanovský and P. Vojtěchovský, Commutator theory for loops, Journal of Algebra, 399 (2014), 290–322.
[24] P. Syrbu and I. Grecu, On total inner mapping groups of middle Bol loops, In Proceedings of the Fifth Conference of Mathematical Society of the Republic of Moldova, September 28–October 1, Chisinau, 2 (2019), 161–164.
[25] P. Syrbu, On a generalization of the inner mapping group, in Proceedings of the Fourth Conference of Mathematical Society of Moldova, June 28–July 2, Chisinau, 2017, pp. 161–164.
[26] A. R. T. Solarin, J. O. Adeniran, T. G. Jaiyéọlá, A. O. Isere and Y. T. Oyebo, Some varieties of loops: Bol–Moufang and non-Bol–Moufang types, In: Hounkonnou, M. N., Mitrović, M., Abbas, M., Khan, M. (eds),Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture,
Mathematics & Health,Springer, Cham, 2023.
[27] P. Vojtěchovský, Three lectures on automorphic loops, Quasigroups and Related Systems, (1) 23 (2015), 129–163.
Journal of Algebra and Related Topics
Volume 14, Issue 1
August 2026
Pages 159-182

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