On certain isotopic maps of central loops
DOI:
https://doi.org/10.4067/S0716-09172011000300003Keywords:
Central loop, Isotopism, Autotopism, Bryant-Schneider group.Abstract
It is shown that the Holomorph of a C-loop is a C-loop if each element of the automorphism group of the loops is left nuclear. Condition under which an element of the Bryant-Schneider group of a C-loop will form an automorphism is established. It is proved that elements of the Bryant-Schneider group of a C-loop can be expressed a product ofpseudo-automorphisms and right translations ofelements of the nucleus of the loop. The Bryant-Schneider group of a C-loop is also shown to be a kind of generalized holomorph of the loop.
References
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[19] J. D. Phillips and P. Vojtechovsky, The varieties of quasigroups of Bol-Moufang type : An equational reasoning approach J. Alg., 293, pp. 17-33, (2005).
[20] J. D. Phillips and P. Vojtechovsky, C-loops ; An Introduction, Publ. Math. Debrecen, 68(1-2), pp. 115-137, (2006).
[21] Robinson, D.A. The Bryant-Schneider group of a loop, Ann. de la Soc. Sci. de Bruxelles, 94, pp. 69-81, (1980).
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[24] D.A Robinson, A Special Embedding of Bol-Loops in Groups., Acta Math. Hungaricae., 18, pp. 95-113, (1977).
[25] A. R. T. Solarin, On the identities of Bol-Moufang type, Koungpook Math. J., 28(1), pp. 51—62, (1998).
[2] Bryant, B.F. & Schneider, H. Principal loop-isotopes of quasigroups, Can. Jour. Math., 18, pp. 120-125, (1966).
[3] R. H. Bruck,Contribution to the Theory of Loops.,Trans. Amer. Math. Soc., 55, pp. 245-354, (1946).
[4] R. H. Bruck, A Survey of Binary Systems., Springer-Verlag, Berlin-Gottinge-Heidelberg., (1966).
[5] O. Chein, A short note on supernuclear (central) elements of inverse property loops, Arch. Math., 33, pp. 131—132, (1979).
[6] V. O. Chiboka, The Study of Properties and Construction of Certain Finite Order G-loops., Ph. D. Thesis (1990), Obafemi Awolowo University, Ile-Ife, Nigeria., 127pp.
[7] V. O. Chiboka and A.R.T. Solarin, Holomorphs of Conjugacy Closed Loops., Scientific Annals of ”AL.I.CUZA”., 38, pp. 277-283, (1991).
[8] V. O. Chiboka and A.R.T. Solarin, Autotopism Characterization of G-Loops., An. Stiin¸. Univ. ”AL.I.Cuza”. t Iasi. Mat., 39, pp. 19-26, (1993).
[9] F. Fenyves, Extra Loops I, Publ. Math. Debrecen, 15, pp. 235—238, (1968).
[10] F. Fenyves, Extra Loops II, Publ. Math. Debrecen, 16, pp. 187—192, (1969).
[11] T. G. Jaiyeola, An Isotopic Study of C-loops., M. Sc. Dissertation (2005), University of agriculture, Abeokuta, Nigeria.
[12] M. K. Kinyon, J. D. Phillips and P. Vojtechovsky , C-loops : Extensions and construction, J. Alg. & its Appl. (to appear).
[13] M. K. Kinyon, and K. Kunen, The Structure of Extra Loops., 6(, pp. 1-20, (2007).
[14] K. Kunen, Quasigroups, Loops and Associative Laws., J. Alg. 185, pp. 194-204, (1996).
[15] K. Kunen, Moufang Quasigroups., J. Alg. 183, pp. 231-234, (1996).
[16] H.O. Pflugfelder, Quasigroups and Loops: Introduction., Sigma Series in Pure Math. 7, Heldermann Verlag, Berlin, 147, (1990).
[17] H.O. Pflugfelder, Historical notes on Lopp Theory., Comment. Math. Carolinae., 4, 2, pp. 359-370, (2000).
[18] J. D. Phillips and P. Vojtechovsky, The varieties of loops of BolMoufang type, Alg. Univ., 53(3), pp. 115-137, (2005).
[19] J. D. Phillips and P. Vojtechovsky, The varieties of quasigroups of Bol-Moufang type : An equational reasoning approach J. Alg., 293, pp. 17-33, (2005).
[20] J. D. Phillips and P. Vojtechovsky, C-loops ; An Introduction, Publ. Math. Debrecen, 68(1-2), pp. 115-137, (2006).
[21] Robinson, D.A. The Bryant-Schneider group of a loop, Ann. de la Soc. Sci. de Bruxelles, 94, pp. 69-81, (1980).
[22] V.S. Ramamurthi and A.R.T. Solarin, On Finite RC-loops., Publ. Math. Debrecen., 35, pp. 261-264, (1988).
[23] D.A Robinson, Holomorphy Theory of Extra Loops., Publ. Math. Debrecen., 18, pp. 59-64, (1971).
[24] D.A Robinson, A Special Embedding of Bol-Loops in Groups., Acta Math. Hungaricae., 18, pp. 95-113, (1977).
[25] A. R. T. Solarin, On the identities of Bol-Moufang type, Koungpook Math. J., 28(1), pp. 51—62, (1998).
Published
2011-12-10
How to Cite
[1]
J. O. Adeniran, Y. T. Oyebo, and D. Mohamed, “On certain isotopic maps of central loops”, Proyecciones (Antofagasta, On line), vol. 30, no. 3, pp. 303-318, Dec. 2011.
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