Polar graphs and corresponding involution sets, loops and Steiner triple systems

A 1-factorization (or parallelism) of the complete graph with loops (P, E, ) is called polar if each 1-factor (parallel class) contains exactly one loop and for any three distinct vertices x1, x2, x3, if {x1} and {x2, x3} belong to a 1-factor then the same holds for any permutation of the set {1, 2, 3}. To a polar graph (P, E,|| ) there corresponds a polar involution set (P, I), an idempotent totally symmetric quasigroup (P, ∗), a commutative, weak inverse property loop (P,+) of exponent 3 and a Steiner triple system (P, B). We have: (P, E,|| ) satisfies the trapezium axiom ⇔ ∀α ∈ I : αIα = I ⇔(P, ∗) is self-distributive ⇔ (P,+) is a Moufang loop ⇔ (P, B) is an affine triple system; and: (P, E,|| ) satisfies the quadrangle axiom⇔ I3 =I ⇔(P, +) is a group ⇔ (P, B) is an affine space.