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

In this paper we present a technique for building a new loop starting from the loops (K,+), ${(P,\widehat{+})}$ and (P,?+) fulfilling suitable conditions, generalizing the construction presented in Zizioli (J Geom 95(1?C2):173?C186, 2009) where ${K=\mathbb{Z}_2}$ or ${K=\mathbb{Z}_3}$ 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.