Semidirect Product of Loops and Fibrations

Starting from two loops (H,+) and (K,\cdot), a new loop L can be defined by means of a suitable map $ \Theta : K \rightarrow {\rm Sym} \ H $. Such a loop is called {\it semidirect product of H and K with respect to $\Theta$ } and denoted by $ H \times_{\Theta} K =: L$. Here we consider the class of those semidirect products in which $ \Theta : K \rightarrow {\rm Aut} (H,+)$ is a homomorphism, this situation being quite akin to the group case. Some relevant algebraic properties of the loop L (Bol condition, Moufang etc.) can be inherited from H and K. In the case that K is a group we investigate the possibility of describing L by a partition (or fibration). In this way we propose a generalization of [8] for the non associative case.