Silvia Pianta

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.

Left loops, bipartite graphs with parallelism and bipartite involution sets

We describe a representation of any semiregularleft loop by means of asemiregular bipartite involution set or, equivalently, a 1-factorization (i.e., a parallelism) of a bipartite graph, with at least one transitive vertex.In these correspondences,Bol loops are associated on one hand toinvariant regular bipartite involution sets and, on the other hand, totrapezium complete bipartite graphs with parallelism; K-loops (or Bruck loops) are further characterized by a sort of local Pascal configuration in the related graph.

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.

On a class of point-reflection geometries

Manara and Marchi have recently given a system of axioms for point-reflection geometries. In this note classes of kinematic spaces are considered in which such reflection geometries can be found. It will be shown that with an additional axiom they can be described by the core in the sense of Bruck (1971) of commutative kinematic spaces without involutory elements.

On automorphisms for some fibered incidence groups

SummaryIn any kinematic space (with non trivial fibration) the group of all automorphisms of the geometric structure which preserve both parallelisms is shown to consist exactly of those automorphisms of the algebraic structure which preserve the fibration. Moreover a characterization of such group is given for a particular class of kinematic spaces.

Binary operations derived from symmetric permutation sets and applications to absolute geometry

A permutation set (P,A) is said symmetric if for any two elements a,b@?P there is exactly one permutation in A switching a and b. We show two distinct techniques to derive an algebraic structure from a given symmetric permutation set and in each case we determine the conditions to be fulfilled by the permutation set in order to get a left loop, or even a loop (commutative in one case). We also discover some nice links between the two operations and finally consider some applications of these constructions within absolute geometry, where the role of the symmetric permutation set is played by the regular involution set of point reflections.

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).

Generalized euclidean and elliptic geometries, their connections and automorphism groups

The notion of an elliptic plane given 1975 by K. Sörensen [S1] will be extended to the notion of a “generalized elliptic space”. Each such elliptic space is derivable from a generalized euclidean space in the sense of H.-J. Kroll and K. Sörensen [KS]. For the case that the euclidean resp. elliptic space has the dimension 3 resp. 2 there is a one to one correspondence between these structures and quaternion fields. Each quaternion field of characteristic ≠ 2 defines in a natural way a 4-dimensional euclidean and a 3-dimensional elliptic space. But, in general, we do not obtain in this way all 4- resp. 3-dimensional geometries. The geometries derivable from quaternion fields will be characterized. Both of these two classes of geometries are provided with different structures, so that there are different automorphism groups, which will be studied.

On commutativity in point-reflection geometries

To any spatial point-reflection geometry there corresponds a determined commutative kinematic space.

Hyperstructures and linear spaces with parallelism

We define a new class of hyperstructures called ‘double hypergroupoids’ suitable to describe incidence spaces with parallelism and relate geometrical configurations to algebraic properties.