Torsors and Ternary Moufang Loops Arising in Projective Geometry

Abstract

A projective space gives rise to an affine space $V_a$ by taking out a hyperplane $a$. We define a natural ternary product on the set $U_{ab} = V_a \cap V_b$, for any pair $(a,b)$ of hyperplanes. If the space is Desarguesian, we show that this ternary product is para-associative and that it coincides with the torsor structures considered in preceding work by the authors. Compared with that work, it is remarkable that—in the case of a projective space—the torsor structure can be expressed solely in terms of the lattice structure of the geometry. For general projective planes, our construction is closely related to the classical construction of ternary rings associated to such planes. In particular, for Moufang planes we show that $U_{ab}$ is a ternary Moufang loop.