Exploring‎ ‎the ‎Algebraic‎ ‎Properties‎ ‎of Gyrosemigroups and a Characterization of Gyrosemigroups of Order 2

Document Type : Original Scientific Paper

Authors

‎Department of Mathematics, Payame Noor University‎, ‎P.O.Box 19395-3697,Tehran‎, ‎I‎. ‎R‎. ‎Iran

Abstract

‎A significant development in the field of gyrogroups was the introduction of the space of all relativistically admissible velocities‎, ‎which brought gyrogroups into the mainstream‎. ‎A group has various generalizations‎, ‎one of which is the notion of gyrogroups‎. ‎Moreover‎, ‎for any pair (a‎, ‎b) in this structure‎, ‎there exists an automorphism gyr[a‎, ‎b] that fulfills left associativity and left loop property‎. ‎The motivation behind this study is to generalize gyrogroups and semigroups‎, ‎which has led to the introduction of gyrosemigroups‎. ‎Accordingly‎, ‎in this paper‎, ‎some classes of gyrosemigroups are presented‎. ‎Also‎, ‎all gyrosemigroups of order 2 are characterized‎. ‎Furthermore‎, ‎the gyrosemigroups with an identity or a zero are studied‎.

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References

[1] A. A. Ungar, Thomas rotation and parametrization of the Lorentz transformation group, Found. Phys. Lett. 1 (1988) 57 - 89, https://doi.org/10.1007/BF00661317.
[2] A. A. Ungar, Thomas precession and its associated grouplike structure, Am. J. Phys. 59 (1991) 824 - 834, https://doi.org/10.1119/1.16730.
[3] A. A. Ungar, Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction, World Scienti c Publishing Co. Pte. Ltd., Singapore, (2010).
[4] F. Chatelin, Qualitative Computing: A Computational Journey into Nonlin-earity, World Scienti c Publishing, 2012.
[5] M. Ferreira, Factorizations of Mobius gyrogroups, Adv. Appl. Cli ord Algebras 19 (2009) 303 - 323, https://doi.org/10.1007/s00006-009-0154-7.
[6] M. Ferreira, Gyrogroups in Projective Hyperbolic Cli ord Analysis, In: I. Sabadini and F. Sommen (eds) Hypercomplex Analysis and Applications, Trends Math. Springer, Basel, 2011, https://doi.org/10.1007/978-3-0346-0246-4 5.
[7] M. Ferreira and G. Ren, Mobius gyrogroups: A Cli ord algebra approach, J. Algebra 328 (2011) 230-253, https://doi.org/10.1016/j.jalgebra.2010.05.014.
[8] T. Suksumran and K. Wiboonton, Einstein gyrogroup as a B-loop, Rep. Math. Phys. 76 (2015) 63 - 74, https://doi.org/10.1016/S0034-4877(15)30019-7.
[9] J. Lawson, Cli ord algebras, Mobius transformations, Vahlen matrices, and B-loops, Comment. Math. Univ. Carolin. 51 (2010) 319 - 331.
[10] T. Suksumran and A. A. Ungar, Bi-Gyrogroup: the group-like structure induced by Bi-decomposition of groups, Math. Interdisc. Res. 1 (2016) 111-142, https://doi.org/10.22052/MIR.2016.13911.
[11] A. R. Ashra , K. Mavaddat Nezhaad and M. A. Salahshour, Classi cation of gyrogroups of orders at most 31, AUT J. Math. Comput. 5 (2024) 11 - 18, https://doi.org/10.22060/AJMC.2023.21939.1125.
[12] S. Mahdavi, A. R. Ashra  and M. A. Salahshour, Construction of new gyrogroups and the structure of their subgyrogroups, Alg. Struc. Appl. 8 (2021) 17 - 30, https://doi.org/10.22034/AS.2021.1971.
[13] A. H. Cli ord and G. B. Preston, The algebraic theory of semigroups, part 1, Amer. Math. Soc. Surveys 7 (1961) p. 1967.
[14] A. H. Cli ord and G. B. Preston, The algebraic theory of semigroups, Mathematical Surveys, vol. II, Amer. Math. Soc., Providence, RI, 1967.
[15] J. M. Howie, An Introduction to Semigroup Theory, London Mathematical Society monographs, Academic Press, London, 1976.
Mathematics Interdisciplinary Research
Volume 9, Issue 3
September 2024
Pages 255-267

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