Binary operations derived from symmetric permutation sets and applications to absolute geometry

A permutation set (P,A)(P,A) is said symmetric if for any two elements a,b∈Pa,b∈P there is exactly one permutation in A switching a and b. We show two distinct techniques to derive an algebraic structure from a given symmetric permutation set and in each case we determine the conditions to be fulfilled by the permutation set in order to get a left loop, or even a loop (commutative in one case). We also discover some nice links between the two operations and finally consider some applications of these constructions within absolute geometry, where the role of the symmetric permutation set is played by the regular involution set of point reflections.