Loops with two-sided inverses constructed by a class of regular permutation sets

Abstract

In this paper we present a technique for building a new loop starting from the loops (K, +), (P, +) and (P, +) fulfilling suitable conditions, generalizing the construction presented in Zizioli (J Geom 95(1-2):173-186, 2009) where K = Z2 or K = Z3 and (P, +) is an abelian group. We investigate the dependence of the properties of the new loop on the corresponding properties of the initial ones (associativity, Bol condition, automorphic inverse property, Moufang condition), and we provide some examples.Mathematics Subject Classification (2010). Primary 20N05.

The limit rotation loop of a hyperbolic plane

“…This paper is placed in the stream of investigation aiming at describing the relationships between some algebraic structures, such as loops and regular permutation sets, and geometric structures, as hyperbolic geometry and graphs (see e.g. [2,8,4,14,13]).…”