Journal ArticleDOI
Loops with two-sided inverses constructed by a class of regular permutation sets
Stefano Pasotti,Elena Zizioli +1 more
TLDR
In this paper, the authors presented a technique for building a new loop starting from the loops (K,+), (P,\widehat{+})} and satisfying suitable conditions, generalizing the construction presented in Zizioli.Abstract:
In this paper we present a technique for building a new loop starting from the loops (K,+), $${(P,\widehat{+})}$$
and (P, +) fulfilling suitable conditions, generalizing the construction presented in Zizioli (J Geom 95(1–2):173–186, 2009) where $${K=\mathbb{Z}_2}$$
or $${K=\mathbb{Z}_3}$$
and (P, +) is an abelian group. We investigate the dependence of the properties of the new loop on the corresponding properties of the initial ones (associativity, Bol condition, automorphic inverse property, Moufang condition), and we provide some examples.read more
Citations
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Journal ArticleDOI
Loops, regular permutation sets and colourings of directed graphs
Stefano Pasotti,Elena Zizioli +1 more
TL;DR: 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.
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 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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Book
Theory of K-loops
TL;DR: In this paper, the left inverse property and Kikkawa loops are combined with Frobenius Ggroups with Mmany Involutions and K-Loops with Fibrations.
Journal ArticleDOI
Recent developments on absolute geometries and algebraization by K-loops
TL;DR: A spatial version of Pasch's assertion is proved, with that a short proof is given for the fact that (P,L) is an exchange space and Ruoff's proportionality Theorem 8.4 for hyperbolic spaces is presented.
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Combinatorial structures in loops I. Elements of the decomposition theory
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