Journal ArticleDOI
Loops, regular permutation sets and colourings of directed graphs
Stefano Pasotti,Elena Zizioli +1 more
TLDR
In this article, the authors established a correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring and characterized regular permutations sets and colored graphs giving rise to the same loop, to isomorphic loops and to isotopic loops.Abstract:
We establish a correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring and characterize regular permutation sets and, respectively, colored graphs giving rise to the same loop, to isomorphic loops and to isotopic loops.read more
Citations
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Journal ArticleDOI
Slid Product of Loops: a Generalization
Stefano Pasotti,Elena Zizioli +1 more
TL;DR: In this article, a sliding product construction for loops with inverses of Pasotti and Zizioli is presented, starting from loops (K, +) equipped with a well ordering.
Journal ArticleDOI
The limit rotation loop of a hyperbolic plane
Silvia Pianta,Stefano Pasotti +1 more
TL;DR: In this paper, both an algebraic and a geometric representation of the limit rotation loop is presented, and its main properties are analyzed in these settings and its automorphism group is determined.
Journal ArticleDOI
Loops, regular permutation sets and graph colourings
Stefano Pasotti,Elena Zizioli +1 more
TL;DR: A correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring is established and some algebraic properties of the loop are related to configurations of the associated graph.
Journal ArticleDOI
A Geometric Environment for Building up Loops
TL;DR: In this paper, the Klein model of a hyperbolic plane over a Euclidean field is used to obtain regular subsets of direct motions as transversals of the coset space.
Journal ArticleDOI
A construction of loops by means of regular permutation sets
Stefano Pasotti,Elena Zizioli +1 more
TL;DR: This paper investigates the nuclei of L and the normality of subloops isomorphic to ( K , + ) and ( P , + ˆ ) and presents here the general setting ensuring the construction of a new loop starting from loops possessing suitable properties.
References
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Journal ArticleDOI
One-factorizations of the complete graph—A survey
Eric Mendelsohn,Alexander Rosa +1 more
TL;DR: This work surveys known results on the existence and enumeration of many kinds of 1-factorizations of the complete graph and discusses some related questions and topics.
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Loops, reflection structures and graphs with parallelism
TL;DR: In this paper, the correspondence between right loops (P, +) with the property (*) ∀a, b ∈ P : a − (a − b) − b and reflection structures described in [4] is extended to the class of graphs with parallelism.
Journal ArticleDOI
Connections between loops of exponent 2, reflection structures and complete graphs with parallelism
TL;DR: In this article, the complete graph Γ with set of vertices L can be endowed with a parallelism ∥ between edges, where L+ ≔ {a+¦a ∈ L} is a reflection structure in the sense of satisfying additional conditions.
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Polar graphs and corresponding involution sets, loops and Steiner triple systems
TL;DR: A 1-factorization (or parallelism) of the complete graph with loops is called polar as mentioned in this paper, and the same holds for any permutation of the set {1, 2, 3}.