Elena Zizioli

Loops, Reflection Structures and Graphs with Parallelism

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 (P, ε, ∥). In this connection K-loops correspond with trapezium graphs, i.e. complete graphs with parallelism satisfying two axioms (T1), (T2) (cf. §3 ). Moreover (P, ε, ∥ +) is a structure loop (i.e. for each a ∈ P the map a+ : P → P; x → a + x is an automorphism of the graph with parallelism (P, ε, ∥)) if and only if (P, +) is a K-loop or equivalently if (P, ε, ∥) is a trapezium graph.

Connections between loops of exponent 2, reflection structures and complete graphs with parallelism

In a loop (L, +) each a ∈ L defines a permutation a+: L → L; x → a + x. Here (L, +) is called of exponent 2 if a+ o a+ = id for all a ∈ L. Then L+ ≔ {a+¦a ∈ L} is a reflection structure in the sense of [3] satisfying additional conditions and the complete graph Γ with set of vertices L can be endowed with a parallelism ∥ between edges.The relations between the structures (L, +), L+ and the complete graph with parallelism Γ are investigated if further conditions are assumed for one of these structures. So for “+” is assumed that (L, +) is a K-loop or a group.

From involution sets, graphs and loops to loop-nearrings

This is a general frame for a theory which connects the areas of loops, involution sets and graphs with parallelism. Our main results are stated in §5, §6 and §7. In §5 we derive a partial binary operation from an involution set and we discuss if such operation is a Bol operation or a K-operation, in §6, we relate involution sets with loops. In §7 we look for the possibility to construct loop-nearrings by considering the automorphism groups of loops.

K-loops derived from Frobenius groups

We consider a generalization of the representation of the so-called co-Minkowski plane (due to H. and R. Struve) to an abelian group (V,+) and a commutative subgroup G of Aut(V,+). If P = G × V satisfies suitable conditions then an invariant reflection structure (in the sense of Karzel (Discrete Math. 208/209 (1999) 387-409)) can be introduced in P which carries the algebraic structure of K-loop on P (cf. Theorem 1). We investigate the properties of the K-loop (P,+) and its connection with the semi-direct product of V and G. If G is a fixed point free automorphism group then it is possible to introduce in (P,+) an incidence bundle in such a way that the K-loop (P,+) becomes an incidence fibered loop (in the sense of Zizioli (J. Geom. 30 (1987) 144-151)) (cf. Theorem 3).

A Class Of Fibered Loops

SummaryIn this note we introduce a constructive method to obtain fibered loops either of any order n ≥ 4 or of countable order. Moreover we investigate some properties of this class of loops.

Loops, regular permutation sets and graph colourings

Abstract We establish a correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring and relate some algebraic properties of the loop to configurations of the associated graph.

Extension of a class of fibered loops to kinematic spaces

AbstractThe notion of the split extension of a commutative kinematic space is extended to the case of a weak K-loop with an incidence fibration (F, +,

A construction of loops by means of regular permutation sets

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