On a class of power associative LCC-loops

O.O. George ^[1] ; Johnson O. Olaleru ^[1] ; J. O. Adéniran ^[3] ; T. G. Jaiyéolá ^[2]
1. [1] University of Lagos
  
  University of Lagos
  
  Nigeria
2. [2] Obafemi Awolowo University
  
  Obafemi Awolowo University
  
  Nigeria
3. [3] Department of Mathematics, Federal University of Agriculture Abeokuta 110101, Nigeria
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Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 37, Nº 2, 2022, págs. 185-194
Idioma: inglés
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Resumen
- 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.