A class of fibered loops related to general hyperbolic planes

Abstract

In this paper we introduce a class of left conjugacy closed loops which are also fibered in subsemigroups. We inspect the possibility to extend the semigroups of the fibration to commutative subgroups. Then we construct an example of such loops arising from a suitable selected subset of the set of all limit rotations of the hyperbolic plane over an euclidean field

The limit rotation loop of a hyperbolic plane

“…In [7] H. Karzel and the two authors introduced a class of fibered loops arising from a suitable subset of the set of limit rotations of a hyperbolic plane via direct loop derivation in the following way. Let (H, L, α, ≡) be a hyperbolic plane over an euclidean field K and let E be the sets of all ends (see [3, § 27]).…”

“…In the following we will denote by Λ the set of all non-trivial limit rotations of the plane. It is well known that for any pair of distinct points (a, b) ∈ H 2 there exist two limit rotations mapping a to b, thus in [7] an orientation is introduced on Λ in order to select a regular subset employing the notion of cyclic order. Recall that if we denote by E 3 the set of triples of distinct elements of E, a cyclic order on the set of ends of a hyperbolic plane (see also [6]) is a map ζ :…”

“…In particular, in the seminal paper [9], the notion of K-loop is used to provide an algebraic representation of a general hyperbolic geometry over an euclidean field K, employing a suitable set of motions, namely the point reflections, and [5] provides a sort of coordinatization of a general absolute plane. More recently in [7] the authors together with H. Karzel introduced a class of left conjugacy closed loops which admit a fibration consisting of commutative subsemigroups. A motivating example of such loops arises naturally from a selected subset of limit rotations acting regularly on the point-set H of a general hyperbolic plane over a Euclidean field K. For the so derived loop (H, ⊕) it turns out that the left multiplication group is the proper motion group of the plane, isomorphic to P SL 2 (K) (cf.…”

“…More in details in section 2 we briefly recall the main definitions concerning loops, regular permutation sets and hyperbolic geometry and we summarize the construction of the limit rotation loop as presented in [7].…”

“…Section 3 is devoted to present the algebraic model of the limit rotation loop, thus the group P SL 2 (K) is embedded into the 3-dimensional projective space PG(3, K) and the set of limit rotations is identified with a cone in such space. Moreover on the points of this cone an orientation is introduced, which turns out to coincide with that introduced in [7]. Finally the operation of the loop is described in terms of the matrices representing limit rotations.…”