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.

Karzel, H., Pianta, S., Zizioli, E., Polar graphs and corresponding involution sets, loops and Steiner triple systems, <<RESULTS IN MATHEMATICS>>, 2006; (49): 149-160. [doi:10.1007/s00025-006-0214-4] [http://hdl.handle.net/10807/55454]

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

#### Abstract

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.
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Karzel, H., Pianta, S., Zizioli, E., Polar graphs and corresponding involution sets, loops and Steiner triple systems, <<RESULTS IN MATHEMATICS>>, 2006; (49): 149-160. [doi:10.1007/s00025-006-0214-4] [http://hdl.handle.net/10807/55454]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10807/55454`
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