On a class of power associative LCC-loops
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Abstract
Let LWPC denote the identity (xy · x) · xz = x((yx · x)z), and RWPC the mirror identity. Phillips proved that a loop satisfies LWPC and RWPC if and only if it is a WIP PACC loop. Here, it is proved that a loop Q fulfils LWPC if and only if it is a left conjugacy closed (LCC) loop that fulfils the identity (xy · x)x = x(yx · x). Similarly, RWPC is equivalent to RCC and x(x · yx) = (x · xy)x. If a loop satisfies LWPC or RWPC, then it is power associative (PA). The smallest nonassociative LWPC-loop was found to be unique and of order 6 while there are exactly 6 nonassociative LWPC-loops of order 8 up to isomorphism. Methods of construction of nonassociative LWPC-loops were developed.
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How to Cite
George, O., Olaleru, J., Adénı́ranJ., & JaiyéoláT. (2022). On a class of power associative LCC-loops. Extracta Mathematicae, 37(2), 185-194. https://doi.org/10.17398/2605-5686.37.2.185
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Section
Group Theory
References
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[2] R.P. Burn, Finite Bol loops, Math. Proc. Cambridge Philos. Soc. 84 (3) (1978), 377 – 385.
[3] P. Csörgo, A. Drápal, Left conjugacy closed Loops of nilpotency class two, Results Math. 47 (2005), 242 – 265.
[4] A. Drápal, On left conjugacy closed loops with a nucleus of index two, Abh. Math. Sem. Univ. Hamburg 74 (2004), 205 – 221.
[5] A. Drápal, On extraspecial left conjugacy closed loops, J. Algebra 302 (2006), 771 – 792.
[6] E.G. Goodaire, D.A. Robinson, A class of loops which are isomorphic to all loop isotopes, Canadian J. Math. 34 (1982), 662 – 672.
[7] T.G. Jaiyéolá, “ A Study of New Concepts in Smarandache Quasigroups and Loops ”, InfoLearn (ILQ), Ann Arbor, MI, 2009.
[8] M.K. Kinyon, K. Kunen, Power-associative, conjugacy closed loops, J. Algebra 304 (2006), 671 – 711.
[9] G.P. Nagy, P. Vojtěchovský, The LOOPS Package, Computing with quasigroups and loops in GAP 3.4.1. https://www.gap-system.org/Manuals/pkg/loops/doc/manual.pdf
[10] The GAP Group, GAP - Groups, Algorithms, Programming, Version 4.11.0. http://www.gap-system.org
[11] H.O. Pflugfelder, “ Quasigroups and Loops: Introduction ”, Sigma Series in Pure Mathematics, 7, Heldermann Verlag, Berlin, 1990.
[12] J.D. Phillips, A short basis for the variety of WIP PACC - loops, Quasigroups Related Systems 14 (2006), 73 – 80.