Bryant-Schneider Group of Basarab Loop
Download PDF
Authors: B. OSOBA, T. G. JAIYéOLá AND O. BALOGUN
DOI: 10.46793/KgJMat2608.1297O
Abstract:
A loop (Q,∘) is called Basarab loop if it is both a left and a right Basarab loop; (x∘yxρ) ∘xz = x∘yz and yx∘ (xλz ∘x) = yz ∘x hold for all x,y,z ∈ Q respectively. In this paper, the characterizations of the Bryant-Schneider group of a Basarab loop are studied using the left and right Basarab loop identities. It is shown that the element, xλ (xρ) is in the left (right) nucleus if and only if the middle inner map Tx (inverse Tx−1) is an automorphism. It is revealed that every crypto-automorphism of a Basarab loop is an element of the Bryant-Schneider group. Some related algebraic properties were also characterized. Furthermore, elements of the Bryant-Schneider group of a Basarab loop in terms of pseudo-automorphism and automorphism are also characterized. A subgroup of the Bryant-Schneider group, characterized by the Basarab loop, is established. Finally, a right pseudo-automorphic characterization of the isotopy-isomorphy of a Basarab loop is carried out.
Keywords:
Quasigroup, loop, crypto-automorphism, pseudoautomorphism Bryant-Schneider group, Basarab loops.
References:
[1] J. O. Adeniran, Some properties of the Bryant-Schneider groups of certain Bol loops, Proc. Jangjeon Math. Soc. 6 (1) (2003), 71–80.
[2] B. Cote, B. Harvill, M. Huhn and A. Kirchman, Classification of loops of generalised Bol-Moufang type, Quasigroups Related Systems 19 (2010), 193–206.
[3] R. C. Di Cocco, On isotopism and pseudo-automorphism of the loops, Bollettino dell’Unione Matematica Italiana 7 (1993), 199–205.
[4] A. S. Basarab, Non-associative extentions of groups by means of abelian groups, Proceedings of the Second International Conference on the Theory of Groups, Timishoara, (1992), 6–10.
[5] A. S. Basarab, K-loops, (in Russian), Buletinul de Identitate al Cetăţeanului Republicii Moldova, Ser. Matematica 1(7) (1992), 28–33.
[6] A. S. Basarab, Generalized Moufang G-loops, Quasigroups Related Systems 3 (1996), 1–5.
[7] A. S. Basarab, IK-loops, Quasigroups Related System 4 (1997), 1–7.
[8] G. O. Effiong, T. G. Jaiyéọlá and M. C. Obi, Characterization of some subloops of a Basarab loop, Annals of Mathematics and Computer Science 4 (2021), 28–47.
[9] G. O. Effiong, T. G. Jaiyéọlá, M. C. Obi and L. S. Akinola, Holomorphy of Basarab loops, Gulf J. Math. 17(1) (2024), 142–166. https://doi.org/10.56947/gjom.v17i1.2076.
[10] T. G. Jaíyéọlá, A Study of New Concepts in Smarandache Quasigroups and Loops, ProQuest Information and Learning (ILQ), Ann Arbor, 2009. http://doi.org/10.5281/zenodo.8913
[11] T. G. Jaiyéọlá, Smarandache isotopy of second Smarandache Bol loops, Scientia Magna Journal 7(1) (2011), 82–93. http://doi.org/10.5281/zenodo.234114.
[12] T. G. Jaiyéọlá, B. Osoba and A. Oyem, Isostrophy Bryant-Schneider group-invariant of Bol loops, Bul. Acad. Ştiinţe Repub. Mold. Mat. 9(99) (2022), 3–18. https://doi.org/10.56415/basm.y2022.i2.p3
[13] T. G. Jaiyéọlá, On Smarandache Bryant Schneider group of a Smarandache loop, International Journal of Mathematical Combinatorics 2 (2008) 51–63. http://doi.org/10.5281/zenodo.820935
[14] T. G. Jaiyéọlá, J. O. Adéníran and A. R. T. Sòlárìn, Some necessary conditions for the existence of a finite Osborn loop with trivial nucleus, Algebras, Groups and Geometries 28(4) (2011), 363–380.
[15] T. G. Jaiyéọlá, J. O. Adéníran and A. A. A. Agboola, On the Second Bryant Schneider group of universal Osborn loops, ROMAI J. 9(1) (2013), 37–50.
[16] T. G. Jaiyéọlá and G. O. Effiong, Basarab loop and its variance with inverse properties, Quasigroups Related Systems 26(2) (2018), 229–238.
[17] T. G. Jaiyéọlá and G. O. Effiong, Basarab loop and the generators of its total multiplication group, Proyecciones 40(4) (2021), 939–962. http://dx.doi.org/10.22199/issn.0717-6279-4430.
[18] B. Osoba and Y. T. Oyebo, On multiplication groups of middle Bol loop related to left Bol loop, International Journal of Mathematics and Its Applications 6(4) (2018), 149–155.
[19] O. Benard and O. Y. Tunde, Isostrophy invariant α-elastic property, Proyecciones 43(6) (2024), 1373–1388.
[20] B. Osoba and Y. T. Oyebo, On Relationship of multiplication groups and isostrophic quasigroups, International Journal of Mathematics Trends and Technology 58(2) (2018), 80–84. http://dx.doi.org/10.14445/22315373/IJMTT-V58P511
[21] B. Osoba, Smarandache nuclei of second Smarandache Bol loops, Scientia Magna Journal 17(1) (2022), 11–21.
[22] B. Osoba and T. G. Jaiyéọlá, Algebraic connections between right and middle Bol loops and their cores, Quasigroups Related Systems 30 (2022), 149–160. https://doi.org/10.56415/qrs.v30.13
[23] B. Osoba and Y. T. Oyebo, On the core of second Smarandach Bol loops, International Journal of Mathematical Combinatorics 2 (2023), 18–30. http://doi.org/10.5281/zenodo.32303
[24] T. Y. Oyebo, B. Osoba, and T. G. Jaiyéọlá, Crypto-automorphism group of some quasigroups, Discuss. Math. Gen. Algebra Appl. 44 (2024), 57–72.
[25] B. Osoba, A. O. Abdulkareem, T. Y Oyebo and A. Oyem, Some characterizations of generalized middle Bol loops, Discuss. Math. Gen. Algebra Appl. (accepted). https://doi.org/10.7151/dmgaa.1457
[26] H. O. Pflugfelder,Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics 7, Heldermann Verlag, Berlin, 1990.
[27] J. D. Phillips and V. A. Shcherbacov, Cheban loops, J. Gen. Lie Theory Appl. 4 (2010), 5 pages. https://doi.org/10.4303/jglta/G100501
[28] D. A. Robinson, The Bryant-Schneider group of a loop, Ann. Soc. Sci. Bruxelles Ser. I 9(4) (1980), 69–81.
[29] 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: M. N. Hounkonnou, M. Mitrović, M. Abbas, M. Khan (Eds.), Algebra without Borders-Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham, 2023, 97–164. https://doi.org/10.1007/978-3-031-39334-1\_3
[30] V. A. Shcherbacov, Elements of Quasigroup Theory and Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2017.
[31] U. Vishne, Four halves of the inverse property in loop extensions, Quasigroups Related Systems 29 (2021), 283–302.
Login:
