ALGEBRAIC PROPERTIES OF MONOGENIC SOFT QUASIGROUPS
Keywords:
Quasigroups, Soft sets, soft quasigroups, soft subquasigroups, monogenic quasigroup
Abstract
This paper explores the properties of soft monogenic quasigroups, a class of algebraic structures that combine soft set theory and quasigroups. We define soft monogenic quasigroups and investigate their generation criteria, structural properties, and invariance under automorphism. Our results contribute to the advancement of soft algebraic structures, with potential applications in fields like cryptography and coding theory. We establish key characteristics of soft monogenic quasigroups, including their generation by a single element and structural invariance. This work lays the foundation for further research in soft quasigroups and their applications.
References
[1] S. O. Ajala, J. O. Aderohunmu, S. O. Ogunrinade, A. Oyem. Bruck Loop and its paras- trophs, Pioneer Journal of Algebra, Number Theory and its Applications. 5(2,) 47 - 54.(2013).
[2] H. Aktas, N. Cagman. Soft Sets and Soft Groups, Inf. Sci. 177, 2726 - 2735, (2007).
[3] H. Aktas, S. Ozlu. Cyclic soft groups and their applications on Groups, The Scientific World Journal. 2014 Article Id 437324, 5 (2014).
[4] A. Albert, A. Baer. Quasigroups, II. Trans. Amer. Math. Soc. 55, 401 - 419, (1944).
[5] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabira. On some new operations in soft set theory, Computers and Mathematics with Applications. 57 (9) 15471553 (2009)
[6] S. Aslihan, O. Atagun. Soft groups and Normalistic soft groups. Computer and Maths with application. 62(2) 685 - 698 (2011).
[7] G. Bol. Uber Drieri - Gewebe im vierdiemensionalen Raun, Math Ann. 110, 431 - 463 (1935).
[8] R. H. Bruck. Contribution to the theory of quasigroups, Trans. Amer. Math. Soc. 60, 245 - 354 (1946).
[9] N. Cagman, E. Enginoglu, Soft matrix theory and its decision making, Computers and Mathematics with Applications 59 33083314 (2010).
[10] D. Chen, C. Tsang, D. Yeung, X. Wang. The parameter reduction of soft sets and its applications, Comput. Math. Appl. 49, 757 - 763 (2005).
[11] O. Chein, O. H. Pflugfelder, J. D. Smith. Quasigroups and Loops: Theory and Applications. Heldermann Verlag (1990).
[12] C. Degang, T. Tsang, W. Xizhao. The Parameterization Reduction of Soft Sets and its Application. Computer and Mathematics with application. 49,757 - 763 (2005).
[13] R. M. Falcon. A new quasigroup isomorphism invariant arising from fractal image patterns Quasigrups and Related Systems. 30(2022), 81 - 90. doi/10.56415/qrs.v30.06
[14] V. Izbash. Monoquasigroups isotopic to loops. Quasigroups and related systems. 3,720, (1996).
[15] D. L. Johnson. Topics in the theory of group presentations , volume 42 of London Mathe- matical Society Lecture Note Series. Cambridge University Press, Cambridge, (K1980).
[16] D. H. S. Jonathan. An Introduction to Quasigroups and their Representations. Chapman and Hall (1999).
[17] Y. Kim and S. H. Kim. On some finite orders of certain cyclically presented groups. Algebra Colloq., 11(4), 477482, (2004).
[18] K. Maji, R. Roy, A., Biswas, R., An Application of Soft Sets in a decision making problem, Comput. Math. Appl. 44, 1077 - 1083 (2002). DOI:10.1016/s0898-1221(02)00216-x
[19] F. Marty. Sur une generalization de la notion de group, in: Proc. 8th Congress Math. Scandenaves, Stockholm, 4549 (1934)
[20] B. D. Mckay, I. M. Wanless X. Zhang. The order of Automorphisms of Quasigroup, Journal of Combinatorial Design (2014).
[21] D. Molodtsov. Soft Set Theory - First Results, Comput. Math. 37, 19 - 31 (1999). doi.org/10.1016/S0898-1221(99)00056-5.
[22] R. Moufang. Zur Struktur von alternativkor, Maths Ann. 110, 416 - 430(1934)
[23] A. Oyem, T. G. Jaiy´eo. l´a,J. O. Olaleru. Order of Finite Soft Quasigroups with Application to Egalitarianism, Discussiones Mathematicae, General Algebra and Applications.42(1):135- 157 (2020). DOI:10.7151/dmgaa.1381.
[24] A. Oyem, J. O. Olaleru, T. G. Jaiy´eo. l´a, H. Akewe. Some algebraic properties of soft quasigroups. International Journal of Math. Sci. and Optimization. 6(2) 834 - 846 (2020). doi:10.6084/m9.figshare.13524392
[25] A. Oyem, T. G. Jaiyeola, Parastrophes and cosets of Soft Quasigroups, International Journal of Mathematical Sciences and Optimization 8(1) 73 - 85 (2022)
DOI:10.48550/arXiv.2207.06582
[26] A. Oyem, J. O. Olaleru, T. G. Jaiyeola, B. Osoba. Soft Neutrosophic Quasigroups, Neu- trosophic Sets and Systems. 50, 488 - 503 (2022) DOI:10.5281/zenodo.6774903
[27] A. Sezgin, A. O. Atagun. On Operations of Soft Sets , Computers and Math with Appli- cation. 61 1457 - 1467 (2011). DOI:10.1016/j.camwa.2011.01.018.
[28] H. O. Pflugfelder. Quasigroups and Loops: Introduction, 160 ISBN 3 - 88538 - 007 - 2.(1999)
[29] Z. Ping, W. Qiaoyan. Operations on Soft Set, Journal of Applied Mathematics. 2013(5), Article Id 105752 (2013). DOI:10.1155/2013/105752.
[30] Z. Qinghau, X. Qin, W. Guoyin. A survey of Rough Sets and their properties. Computer and Maths with Application. 59, 787 - 794 (2016). DOI:10.1016/jtrit.2016.11.001
[2] H. Aktas, N. Cagman. Soft Sets and Soft Groups, Inf. Sci. 177, 2726 - 2735, (2007).
[3] H. Aktas, S. Ozlu. Cyclic soft groups and their applications on Groups, The Scientific World Journal. 2014 Article Id 437324, 5 (2014).
[4] A. Albert, A. Baer. Quasigroups, II. Trans. Amer. Math. Soc. 55, 401 - 419, (1944).
[5] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabira. On some new operations in soft set theory, Computers and Mathematics with Applications. 57 (9) 15471553 (2009)
[6] S. Aslihan, O. Atagun. Soft groups and Normalistic soft groups. Computer and Maths with application. 62(2) 685 - 698 (2011).
[7] G. Bol. Uber Drieri - Gewebe im vierdiemensionalen Raun, Math Ann. 110, 431 - 463 (1935).
[8] R. H. Bruck. Contribution to the theory of quasigroups, Trans. Amer. Math. Soc. 60, 245 - 354 (1946).
[9] N. Cagman, E. Enginoglu, Soft matrix theory and its decision making, Computers and Mathematics with Applications 59 33083314 (2010).
[10] D. Chen, C. Tsang, D. Yeung, X. Wang. The parameter reduction of soft sets and its applications, Comput. Math. Appl. 49, 757 - 763 (2005).
[11] O. Chein, O. H. Pflugfelder, J. D. Smith. Quasigroups and Loops: Theory and Applications. Heldermann Verlag (1990).
[12] C. Degang, T. Tsang, W. Xizhao. The Parameterization Reduction of Soft Sets and its Application. Computer and Mathematics with application. 49,757 - 763 (2005).
[13] R. M. Falcon. A new quasigroup isomorphism invariant arising from fractal image patterns Quasigrups and Related Systems. 30(2022), 81 - 90. doi/10.56415/qrs.v30.06
[14] V. Izbash. Monoquasigroups isotopic to loops. Quasigroups and related systems. 3,720, (1996).
[15] D. L. Johnson. Topics in the theory of group presentations , volume 42 of London Mathe- matical Society Lecture Note Series. Cambridge University Press, Cambridge, (K1980).
[16] D. H. S. Jonathan. An Introduction to Quasigroups and their Representations. Chapman and Hall (1999).
[17] Y. Kim and S. H. Kim. On some finite orders of certain cyclically presented groups. Algebra Colloq., 11(4), 477482, (2004).
[18] K. Maji, R. Roy, A., Biswas, R., An Application of Soft Sets in a decision making problem, Comput. Math. Appl. 44, 1077 - 1083 (2002). DOI:10.1016/s0898-1221(02)00216-x
[19] F. Marty. Sur une generalization de la notion de group, in: Proc. 8th Congress Math. Scandenaves, Stockholm, 4549 (1934)
[20] B. D. Mckay, I. M. Wanless X. Zhang. The order of Automorphisms of Quasigroup, Journal of Combinatorial Design (2014).
[21] D. Molodtsov. Soft Set Theory - First Results, Comput. Math. 37, 19 - 31 (1999). doi.org/10.1016/S0898-1221(99)00056-5.
[22] R. Moufang. Zur Struktur von alternativkor, Maths Ann. 110, 416 - 430(1934)
[23] A. Oyem, T. G. Jaiy´eo. l´a,J. O. Olaleru. Order of Finite Soft Quasigroups with Application to Egalitarianism, Discussiones Mathematicae, General Algebra and Applications.42(1):135- 157 (2020). DOI:10.7151/dmgaa.1381.
[24] A. Oyem, J. O. Olaleru, T. G. Jaiy´eo. l´a, H. Akewe. Some algebraic properties of soft quasigroups. International Journal of Math. Sci. and Optimization. 6(2) 834 - 846 (2020). doi:10.6084/m9.figshare.13524392
[25] A. Oyem, T. G. Jaiyeola, Parastrophes and cosets of Soft Quasigroups, International Journal of Mathematical Sciences and Optimization 8(1) 73 - 85 (2022)
DOI:10.48550/arXiv.2207.06582
[26] A. Oyem, J. O. Olaleru, T. G. Jaiyeola, B. Osoba. Soft Neutrosophic Quasigroups, Neu- trosophic Sets and Systems. 50, 488 - 503 (2022) DOI:10.5281/zenodo.6774903
[27] A. Sezgin, A. O. Atagun. On Operations of Soft Sets , Computers and Math with Appli- cation. 61 1457 - 1467 (2011). DOI:10.1016/j.camwa.2011.01.018.
[28] H. O. Pflugfelder. Quasigroups and Loops: Introduction, 160 ISBN 3 - 88538 - 007 - 2.(1999)
[29] Z. Ping, W. Qiaoyan. Operations on Soft Set, Journal of Applied Mathematics. 2013(5), Article Id 105752 (2013). DOI:10.1155/2013/105752.
[30] Z. Qinghau, X. Qin, W. Guoyin. A survey of Rough Sets and their properties. Computer and Maths with Application. 59, 787 - 794 (2016). DOI:10.1016/jtrit.2016.11.001
Published
2025-12-29
How to Cite
OJO-OROBOSA, V., OYEM, A., OGBEREYIVWE, O., OSOBA, B., & EKEH, K. I. (2025). ALGEBRAIC PROPERTIES OF MONOGENIC SOFT QUASIGROUPS. Unilag Journal of Mathematics and Applications, 5(2), 1 - 13. Retrieved from https://lagjma.unilag.edu.ng/article/view/2793
Section
Articles
Copyright (c) 2025 VERÓNICA OJO-OROBOSA, ANTHONY OYEM, OGHOVESE OGBEREYIVWE, BERNARD OSOBA, KENNETH IKECHUKWU EKEH
This work is licensed under a Creative Commons Attribution 4.0 International License.
