تعداد نشریات | 43 |
تعداد شمارهها | 1,682 |
تعداد مقالات | 13,763 |
تعداد مشاهده مقاله | 32,249,868 |
تعداد دریافت فایل اصل مقاله | 12,757,958 |
Constructions and involutory properties in latin quandles | ||
International Journal of Group Theory | ||
مقاله 2، دوره 13، شماره 1، خرداد 2024، صفحه 1-15 اصل مقاله (420.18 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22108/ijgt.2022.132799.1780 | ||
نویسندگان | ||
Abednego Orobosa Isere* ؛ Julia Ugomma Ezurike | ||
Department of Mathematics, Faculty of Physical Sciences, Ambrose Alli University, Ekpoma, Nigeria. | ||
چکیده | ||
This work studied involutory properties in Latin quandles using methods of quasiqroup theory, and classified latin quandle $Q$ into Left Involutory Property latin Quandle (LIPQ), Right Involutory Property Latin Quandle (RIPQ) and Involutory Property Latin Quandle (IPQ). It investigated a fourth property called the Cross Involutory Property Latin Quandle (CIPQ). The result showed that a latin quandle $Q$ that is a LIPQ and RIPQ is an IPQ. Moreover, it established that the necessary and sufficient conditions for a latin Alexander quandle $Q$ to be a CIPQ is that $b=t^2a +(1-t)(tb+a)$ for all $a,b \in Q$ and $t\in A(Q)$. | ||
کلیدواژهها | ||
Latin quandles؛ LIPQ؛ RIPQ؛ CIPQ؛ IPQ | ||
مراجع | ||
[1] W. Alexander and G. B. Briggs, On types of knotted curves, Ann. of Math. (2), 28 (1926/27) 562–586.
[2] V. G. Bardakov, P. Dey and M. Singh, Automorphism Groups of Quandles Arising from Groups, Monatsh. Math., 184 (2017) 519–530. [3] M. Bonatto and P. Vojtĕchovský, Simply connected latin quandles, J. Knot Theory Ramifications, 27 (2018) 32 pp.
[4] C. Burstin and W. Mayer, Distributive Gruppen von endlicher Ordnung, (German), J. Reine Angew. Math., 160 (1929) 111–130. [5] V. D. Belousov, Fundamentals of the theory of quasiqroups and loops, Nauka, Moska, (1967) (Russian).
[6] R. H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc., 55 (1944) 19–52.
[7] R. H. Bruck, A survey of binary systems, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958 185 pp.
[8] J. Denes and A. D. Keedwell Latin Squares and their Applications, Academia Kiado, Budapest, 1974.
[9] N. N. Didurik and V. A. Shcherbacov, On definition of CI-quasigroup, ROMAI J., 13 (2017) 55–58.
[10] M. Elhamdadi, Distributivity in quandles and quasigroups, Algebra, Geometry and Mathematical Physics , Springer Proceedings in Mathematics and Statistics, 85, Springer-Valag Heidelberg, 2014 325–340. [11] M. Elhamdadi, J. MacQuarrie and R. Restrepo, Automorphism groups of quandles, J. Algebra Appl., 11 (2012) 9 pp. [12] M. Elhamdadi and S. Nelson, Quandles-an introduction to the algebra of knots, Student Mathematical Library, American Mathematical Society, Providence, 74 2015. [13] V. M. Galkin, Left distributive finite order quasigroups, Quasigroups and loops. Mat. Issled., No. 51 (1979) 43–54.
[14] B. Ho and S. Nelson, Matrices and finite quandles, Homology Homotopy Apl., 7 (2005) 197–208.
[15] E. D. Huthnance Jr, A theory of generalised Moufang loops, Ph.D. thesis, Georgia Institute of Technology, (1968).
[16] Indu R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications, 26 (2018) 1–10. [17] A. O. Isere, J. O. Adeniran and T.G. Jaiyeola, Latin Quandles and Applications to Cryptography, Math. Appl., 10 (2021) 37–53. [18] A. O. Isere, A quandle of order 2n and the concept of quandles isomorphism, J. Nigerian Math. Soc., 39 (2020) 155–166. [19] A. O. Isere, J. O. Adeniran and A. R. T. Solarin, Some examples of finite Osborn loops, J. Nigerian Math. Soc., 31 (2012) 91–106. [20] A. O. Isere, S. A. Akinleye and J. O. Adeniran, On Osborn loops of order 4N , Acta Univ. Apulensis Math. Inform., No. 37 (2014) 31–44. [21] A. O. Isere, J. O. Adeniran and T.G. Jaiyeola, Generalized Osborn Loops of Order 4n, Acta Univ. Apulensis Math. Inform., No. 43 (2015) 19–31. [22] A. O. Isere, J. O. Adénı́ran and T. G. Jaiyéolá, Classification of Osborn loops of order 4n, Proyecciones, 38 (2019) 31–47. [23] A. O. Isere, O. A. Elakhe and C. Ugbolo , A Higher Quandle of order 24, and its Inner Automorphisms, J. Physical & Applied Sciences, 1 (2018) 100–110. [24] A. O. Isere, J. O. Adeniran and T.G. Jaiyeola, Holomorphy of Osborn Loops, An. Univ. Vest Timiş. Ser. Mat.- Inform., 53 (2015) 81–98. [25] T G. Jaiyéo.lá and E. Effiong, Basarab loop and its variance with inverse properties, Quasigroups and Related Systems, 26 (2018) 229–238. [26] D. Joyce, A classifying invariant of knots, the Knot Quandle, J. Pure Appl. Algebra, 23 (1982) 37–66. [27] D. Joyce, Simple Quandles, J. Algebra, 79 (1982) 307–318. [28] S. Kamada, Knot invariants derived from quandles and racks, Invariants of knots and 3-manifolds, Geom. Topol. Monogr., 4 (2001) 103–117. [29] S. Kamada, H. Tamaru and K. Wada, On classification of quandles of cycle type, Tokyo. J. Math., 39 (2016) 157–171. [30] A. Krapez, A Note On Belousov quasiqroups, Quasigroups Related Systems, 15 (2007) 291–294.
[31] A. D. Keedwell, Crossed-inverse quasigroups with long inverse cycles and applications to cryptography, Australas. J. Combin., 20 (1999) 241–250. [32] A. D. Keedwell and V. A. Shcherbacov, On m-inverse loops and quasigroups with a long inverse cycle, Australas. J. Combin., 26 (2002) 99–119. [33] A. D. Keedwell and V. A. Shcherbacov, Quasigroups with an inverse property and generalized parastrophic identities, Quasigroups Related Systems, 13 (2005) 109–124. [34] M. K. Kinyon and J. D. Phillips, Axioms for trimedial quasigroups, Comment. Math. Univ. Carolin., 45 (2004) 287–294. [35] J. Macquarrie, Automorphism groups of quandles of orders 3, 4 and 5, Graduate Thesis and Dissertation, University of South Florida available at https://scholarcommons.usf.edu/etd/3226 (2011). [36] S. V. Matveev, Distributive groupoids in knot theory, (Russian), mat. sb. (N. S), 119 (1982) 78–88.
[37] K. McCrimmon, A taste of Jordan algebras, Universitext, Springer-Verlag, New York, 2004.
[38] G. Murillo, S. Nelson and A. Thompson, Matrices and finite Alexander quandles, J. Knot Theory Ramifications, 16 (2007) 769–778. [39] F. Orrin, Symmetric and self-distributive systems, Amer. Math. Monthly, 62 (1955) 699–707.
[40] H. O. Pflugfelder, Quasigroups and loops: Introduction, Sigma series in Pure Math. 7, Heldermann Verlag, Berlin, (1990) 147pp. [41] V. A. Shcherbacov, Elements of quasigroup theory and some its applications in code theory and cryptology, 2003 pp. 85. [42] D. A. Stanovský, A guide to self-distributive quasigroups or latin quandles, Quasigroups Related Systems, 23 (2015) 91–128. [43] J. D. H. Smith, Finite distributive quasigroups, Math. Proc. Cambridge Philos. Soc., 80 (1976) 37–41.
[44] J. D. H. Smith, Lectures On quasigroup Representations, Quasigroups Related Systems, 15 (2007) 109–140.
[45] M. Takasaki, Abstractions of symmetric functions, Tohoku Math. J., 49 (1943) 143–207.
[46] Waterloo Maple Inc, Maple 18 (computer software), Ontario: Waterloo, (2014). | ||
آمار تعداد مشاهده مقاله: 341 تعداد دریافت فایل اصل مقاله: 263 |