Proyecciones Journal of Mathematics Vol. 30, N^o 3, pp. 303-318, December 2011. Universidad Católica del Norte Antofagasta - Chile

 

On certain isotopic maps of central loops

 

John Olusola Adeniran

University of Agriculture, Nigeria

Yacub Tunde Oyebo

Lagos State University, Nigeria

Daabo Mohammed

University for Development Studies, Ghana

 

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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.

AMS Classification : Primary 20NO5; Secondary 08A05.

Keywords : Central loop, isotopism, autotopism, Bryant-Schneider group.

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REFERENCES

[1] J.O.Adeniran, On Some Maps of Conjugacy Closed Loops., An. Stiinf. Univ."AL.I.Cuza". Iasi. Mat. 50(2004), pp.267-272, (2004).

[2] Bryant, B.F. & Schneider, H. Principal loop-isotopes of quasigroups, Can. Jour. Math., 18, pp. 120-125, (1966).         [ Links ]

[3] R.H.Bruck, Contribution to the Theory of Loops., Trans. Amer. Math. Soc., 55, pp. 245-354, (1946).         [ Links ]

[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.         [ Links ]

[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).         [ Links ]

[8] V.O.Chiboka and A.R.T.Solarin, Autotopism Characterization of G-Loops, An. Stiint. Univ. "AL.I.Cuza". Iasi.Mat., 39, pp. 19-26, (1993).         [ Links ]

[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. Vojtechovský , C-loops : Extensions and construction, J. Alg. and 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).         [ Links ]

[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. Vojtechovský, The varieties of loops of Bol-Moufang type, Alg.Univ. , 53(3), pp. 115-137, (2005).         [ Links ]

[19] J. D. Phillips and P. Vojtechovský, 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. Vojtechovský, 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).         [ Links ]

[24] D.A Robinson, A Special Embedding of Bol-Loops in Groups., Acta Math. Hungaricae., 18, pp. 95-113, (1977).         [ Links ]

[25] A. R. T. Solarin, On the identities of Bol-Moufang type, Koungpook Math. J., 28(1), pp. 51—62, (1998).

 

John Olusola Adeniran

Department of Mathematics University of Agriculture

Abeokuta 110101

Nigeria

e-mail : adeniranoj@unaab.edu.ng ; ekenedilichineke@yahoo.com

Yakub Tunde Oyebo

Department of Mathematics Lagos State University

Ojo,

Nigeria

e-mail : oyeboyt@yahoo.com and

Daabo Mohammed

Department of Mathematics University for Development Studies Tamale Ghana

e-mail : daabo2005@yahoo.com

Received : October 2010. Accepted : September 2011