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Polar Graphs and Corresponding Involution Sets, Loops and Steiner Triple Systems

Authors:	Helmut Karzel Silvia Pianta Elena Zizioli

Institution:	(1) Zentrum Mathematik, T.U. München, D-80290 München, Germany;(2) Dipartimento di Matematica e Fisica, Università Cattolica, Via Trieste, 17, I-25121 Brescia, Italy;(3) Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti, 9, I-25133 Brescia, Italy

Abstract:	A 1-factorization (or parallelism) of the complete graph with loops $(P, \mathcal{E}, \|\|)$ 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 $(P, \mathcal{E}, \|\|)$ there corresponds a polar involution set $(P, \mathcal{I})$ , an idempotent totally symmetric quasigroup (P, ), a commutative, weak inverse property loop (P, + ) of exponent 3 and a Steiner triple system $(P, \mathcal{B})$ . We have: $(P, \mathcal{E}, \|\|)$ satisfies the trapezium axiom $\Leftrightarrow \forall \alpha \in \mathcal{I}: \alpha \mathcal{I}\alpha = \mathcal{I} \Leftrightarrow (P,)$ is self-distributive ⇔ (P, + ) is a Moufang loop $\Leftrightarrow (P, \mathcal{B})$ is an affine triple system; and: $(P, \mathcal{E}, \|\|)$ satisfies the quadrangle axiom $\Leftrightarrow \mathcal{I}^{3} = \mathcal{I} \Leftrightarrow (P, + )$ is a group $\Leftrightarrow (P, \mathcal{B})$ is an affine space.

Keywords:	20N05 05C70 51E10
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