Construction of a class of loops via graphs

摘要

Starting from an involutorial difference loop (P, +) of order n weconstruct a new loop (L, ) in the same class and possessing 2n elements.The construction was induced by studying a correspondence connectingsuch loops with complete graphs with parallelism and regular involutionsets (see "Basic notions and known results"). We discuss conditions on(P, +) ensuring that the loop (L, ) is a K-loop and we give explicitexamples of K-loops obtained with this method. Further generalizationsof this technique are proposed as well.