A class of fibered loops
A fibered loop (P,\({\mathcal F},\cdot)\) is a loop (P,\(\cdot)\) together with a set \({\mathcal F}\) of its subloops pairwise intersecting only in the identity element of (P,\(\cdot)\) and such that \(\cup {\mathcal F}=P\). The aim of this paper is to determine fibered loops which are not groups. The author presents a method to construct such structures for any order \(n\geq 4\) and for countable order. The construction stems from a class of semi-diagonal latin squares \([a_{ij}]\) of order m (i.e. such that \(a_{ii}=i\) for any \(i\in \{1,2,...,m\})\) whose matrices are explicitly determined for any order m. Some properties of the so obtained fibered loops are investigated, in particular it turns out that this method gives proper fibered loops for any order \(n>4\) and that such fibered loops (P,\({\mathcal F},\cdot)\) carry a geometric structure of incidence space if and only if every element of \({\mathcal F}\) is a subgroup of order 2.
scientific article; zbMATH DE number 4047972