Nuclear identification of some new loop identities of length five

Olufemi Olakunle George ; Tèmítópé Gbóláhàn Jaíyéolá

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2022), pp. 39-58.

Résumé

In this work, we discovered a dozen of new loop identities we called identities of 'second Bol-Moufang type'. This was achieved by using a generalized and modified nuclear identification model originally introduced by Drápal and Jedlic̆ka. Among these twelve identities, eight of them were found to be distinct (from well known loop identities), among which two pairs axiomatize the weak inverse property power associative conjugacy closed (WIP PACC) loop. The four other new loop identities individually characterize the Moufang identities in loops. Thus, now we have eight loop identities that characterize Moufang loops. We also discovered two (equivalent) identities that describe two varieties of Buchsteiner loops. In all, only the extra identities which the Drápal and Jedlic̆ka nuclear identification model tracked down could not be tracked down by our own nuclear identification model. The dozen laws $\{Q_i{\}}_{i=1}^{12}$ induced by our nuclear identification form four cycles in the following sequential format: $(Q_{4i-j}{)}_{i=1}^3,j=0,1,2,3,$ and also form six pairs of dual identities. With the help of twisted nuclear identification, we discovered six identities of lengths five that describe the abelian group variety and commutative Moufang loop variety (in each case). The second dozen identities $\{Q_i^{\ast }{\}}_{i=1}^{12}$ induced by our twisted nuclear identification were also found to form six pairs of dual identities. Some examples of loops of smallest order that obey non-Moufang laws (which do not necessarily imply the other) among the dozen laws $\{Q_i{\}}_{i=1}^{12}$ were found.

Bibliographie

[1] Adéníran J. O., Jaiyéolá T. G., “On central loops and the central square property”, Quasigroups Relat. Syst., 15:2 (2007), 191–200 | MR

[2] Akhtar R., Arp A., Kaminski M., Van Exel J., Vernon D., Washington C., “The varieties of Bol-Moufang quasigroups defined by a single operation”, Quasigroups Relat. Syst., 20:1 (2012), 1–10 | MR

[3] Beg A., “A theorem on C-loops”, Kyungpook Math. J., 17 (1977), 91–94 | MR

[4] Beg A., “On LC-, RC-, and C-loops”, Kyungpook Math. J., 20 (1980), 211–215 | MR

[5] Bruck R. H., A survey of Binary Systems, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1958 | MR

[6] Chein O., Pflugfelder H. O., Smith J. D. H., Quasigroups and Loops: Theory and Applications, Heldermann Verlag, 1990 | MR

[7] Cote B., Harvill B., Huhn M., Kirchman A., “Classification of loops of Bol-Moufang type”, Quasigroups Relat. Syst., 19 (2011), 193–206 | MR

[8] Csorgo P., Drápal A., Kinyon K., “Buchsteiner Loops”, Int. J. Algebra Comput., 19:8 (2009), 1049–1088 | DOI | MR

[9] Drápal A., Jedlic̆ka P., “On loop identities that can be obtained by a nuclear identification”, Eur. J. of Comb., 31:7 (2010), 1907–1923 | DOI | MR

[10] Drápal A., Kinyon M., “Normality, nuclear squares and Osborn identities”, Commentat. Math. Univ. Carol., 61:4 (2020), 481–500 | MR

[11] Fenyves F., “Extra loops. I”, Publ. Math. Debr., 15 (1968), 235–238 | DOI | MR

[12] Fenyves F., “Extra loops. II: On loops with identities of Bol-Moufang type”, Publ. Math. Debr., 16 (1969), 187–192 | DOI | MR

[13] George O. O., Olaleru J. O., Adéníran J. O., Jaiyéolá T. G., “On a class of power associative LCC-loops”, Extracta Math., 37:2 (2022), 185–194 | DOI | MR

[14] Ilojide E., Jaíyéolá T. G., Olatinwo M. O., “On Holomorphy of Fenyves BCI-Algebras”, J. Niger. Math. Soc., 38:2 (2019), 139–155 | MR

[15] Jaiyéolá T. G., An isotopic study of properties of central loops, M.Sc. dissertation, University of Agriculture, Abeokuta, Nigeria, 2005

[16] Jaiyéolá T. G., “On the universality of central loops”, Acta Univ. Apulensis, Math. Inform., 19 (2009), 113–124 | MR

[17] Jaiyéolá T. G., A study of new concepts in Smarandache quasigroups and loops, InfoLearnQuest (ILQ), Ann Arbor, MI, 2009, 127 pp. | MR

[18] Jaiyéolá T. G., “Generalized right central loops”, Afr. Mat., 26:7-8 (2015), 1427–1442 | DOI | MR

[19] Jaiyéolá T. G., Adéníran J. O., “On the derivatives of central loops”, Adv. Theor. Appl. Math., 1:3 (2006), 233–244 | MR

[20] Jaiyéolá T. G., Adéníran J. O., “Algebraic properties of some varieties of loops”, Quasigroups Relat. Syst., 16:1 (2008), 37–54 | MR

[21] Jaiyéolá T. G., Adéníran J. O., “On some autotopisms of non-Steiner central loops”, J. Niger. Math. Soc., 27 (2008), 53–67 | MR

[22] Jaiyéolá T. G., Adéníran J. O., “On isotopic characterization of central loops”, Creat. Math. Inform., 18:1 (2009), 39–45 | MR

[23] Jaiyéolá T. G., Adéníran J. O., “A new characterization of Osborn-Buchsteiner loops”, Quasigroups Relat. Syst., 20:2 (2012), 233–238 | MR

[24] Jaíyéolá T. G., Adeniregun A. A., Asiru M. A., “Finite FRUTE Loops”, J. Algebra Appl., 16:2 (2017), 10 pp. | DOI | MR

[25] Jaíyéolá T.G., Sòlárìn A. R. T., Adéníran J. O., “Some Bol-Moufang characterization of the Thomas precession of a gyrogroup”, Algebras Groups Geom., 31:3 (2014), 341–362 | MR

[26] Jaíyéolá T. G., Ilojide E., Olatinwo M. O., Smarandache F., “On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)”, Symmetry, 10:10 (2018), 427 pp. | DOI | MR

[27] Jaíyéolá T. G., Ilojide E., Saka A. J., Ilori K. G., “On the Isotopy of some Varieties of Fenyves Quasi Neutrosophic Triplet Loop (Fenyves BCI-algebras)”, Neutrosophic Sets and Systems, 31 (2020), 200–223 | DOI | MR

[28] Jaíyéolá T. G., Adeniregun A. A., Oyebola O. O., Adelakun A. O., “FRUTE loops”, Algebras, Groups and Geometries, 37:2 (2021), 159–179 | DOI

[29] Nagy G. P., Vojtěchovský P., The LOOPS Package, Computing with quasigroups and loops in GAP 3.4.1 http://www.math.du.edu/loops

[30] The GAP Group, GAPS - Groups, Algorithms, Programming, Version 4.11.0, http://www.gap-system.org/Manuals/pkg/loops/doc/manual.pdf

[31] Pflugfelder H. O., Quasigroups and loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann Verlag, Berlin, 1990, 147 pp. | MR

[32] Phillips J. D., “A short basis for the variety of WIP PACC-loops”, Quasigroups Relat. Syst., 14:1 (2006), 73–80 | MR

[33] Osborn J. M., “Loops with the weak inverse property”, Pac. J. Math., 10 (1960), 295–304 | DOI | MR

[34] Phillips J. D., Vojtẽchovský P., “The varieties of quasigroups of Bol-Moufang type: an equational reasoning approach”, J. Algebra, 293:1 (2005), 17–33 | DOI | MR

[35] Phillips J. D., Vojtẽchovský P., “The varieties of loops of Bol-Moufang type”, Algebra Univers., 54:3 (2005), 259–271 | DOI | MR

[36] Phillips J. D., Vojtẽchovský P., “C-loops: an introduction”, Publ. Math. Debr., 68:1-2 (2006), 115–137 | DOI | MR

[37] Kinyon M. K., Phillips J. D., Vojtẽchovský P., “C-loops: extensions and constructions”, J. Algebra Appl., 6:1 (2007), 1–20 | DOI | MR

[38] Ramamurthi V. S., Sòlárìn A. R. T., “On finite right central loops”, Publ. Math. Debr., 35:3-4 (1988), 261–264 | DOI | MR

[39] Robinson D. A., “Bol loops”, Trans. Am. Math. Soc., 123 (1966), 341–354 | DOI | MR

[40] Sòlárìn A. R. T., “On the Identities of Bol Moufang Type”, Kyungpook Math. J., 28:1 (1988), 51–62 | MR