From Involution Sets, Graphs and Loops to Loop-Nearrings

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 ⇔ I 3 = I ⇔ (P, +) is a group ⇔ (P, B) is an affine space. Mathematics Subject Classification (2000). 20N05, 05C70, 51E10.

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“…From the definitions one can deduce the following correspondences among the previous structures (see f.i. [7,8]):…”

Section: Notations and Known Resultsmentioning

confidence: 99%

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

Karzel¹,

Pianta

Zizioli

2006

Result. Math.

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“…Let P (2) Then C is an involutory antiautomorphism of (P, ) mapping each chain onto a chain and if D ∈ C is a further chain then the product C • D is 348 H. Karzel et al Results. Math. an automorphism of (P, [1][2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Symmetric 2-Structures, a Classification

Karzel¹,

Kosiorek

Matraś

2013

Results. Math.

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“…From 2000 on a more general class of loops were investigated by looking at a set I of involutions acting transitively on a set P . In particular it has been proved (cf [2,3]) that any regular involution set (P, I), together with the choice of a point o ∈ P , gives rise on the one hand to an involutorial difference loop (P, +), briefly ID-loop and on the other hand to a complete graph (with loops) with vertex set P and possessing a minimal edge-coloring also called parallelism.…”

Section: Introductionmentioning

confidence: 99%

“…[2][3][4][5][6]). The representation of these loops by means of colored graphs furnishes a new approach and also a different point of view in the investigation of such structures.…”

Section: Introductionmentioning

confidence: 99%

Construction of a class of loops via graphs

Zizioli

2009

J. Geom.

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Starting from an involutorial difference loop (P,+) of order n we construct a new loop (L,.) in the same class and possessing 2n elements. The construction was induced by studying a correspondence connecting such loops with complete graphs with parallelism and regular involution sets. We discuss conditions on (P,+) ensuring that the loop (L,.) is a K-loop and we give explicit examples of K-loops obtained with this method. Further generalizations of this technique are proposed as well

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From Involution Sets, Graphs and Loops to Loop-Nearrings

Cited by 5 publications

References 8 publications

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

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

Symmetric 2-Structures, a Classification

Construction of a class of loops via graphs

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