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.

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.

(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.

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.

pp.

[12] M. Elhamdadi and S. Nelson, Quandles-an introduction to the algebra of knots, Student Mathematical Library,

American Mathematical Society, Providence, 74 2015.

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.

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.

(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.

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.

(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.

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.

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.

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.

& 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.

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.

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.

[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.

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.

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.

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.

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.

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.

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).

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.

(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.

(1990) 147pp.

[41] V. A. Shcherbacov, Elements of quasigroup theory and some its applications in code theory and cryptology, 2003

pp. 85.

pp. 85.

[42] D. A. Stanovský, A guide to self-distributive quasigroups or latin quandles, Quasigroups Related Systems, 23 (2015)

91–128.

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).

March 2024

Pages 1-15

**Receive Date:**21 February 2022**Revise Date:**11 July 2022**Accept Date:**13 July 2022**Published Online:**01 March 2024