Constructions and involutory properties in latin quandles

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Physical Sciences, Ambrose Alli University, Ekpoma, Nigeria.

Abstract

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 $t\in A(Q)$.

Keywords

Main Subjects


[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).