1,720,985 research outputs found
Left loops, bipartite graphs with parallelism and bipartite involution sets
No full textWe describe a representation of any semiregular left loop by means of a semiregular 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 to invariant regular bipartite involution sets and, on the other hand, to trapezium complete bipartite graphs with parallelism; K-loops (or Bruck loops) are further characterized by a sort of local Pascal configuration in the related graph
Bynary operations derived from symmetric permutation sets and applications to absolute geometry
No full textA permutation set (P,A) is said symmetric if for any two elements a,b in 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
Absolute planes with elliptic congruence
No full textWe construct examples of non-archimedean absolute planes which are enbedded in elliptic planes, so that they admit an elliptic congruence
Legendre-like theorems in a general absolute geometry.
No full textIn this paper the axiomatic basis will be a general absolute plane
A = (P,L, α,≡) in the sense of [6], where P and L denote respectively the
set of points and the set of lines, α the order structure and ≡ the congruence,
and where furthermore the word “general” means that no claim is made on
any kind of continuity assumptions. Starting from the classification of general
absolute geometries introduced in [5] by means of the notion of congruence,
singular or hyperbolic or elliptic, we get now a complete characterization of
the different possibilities which can occur in a general absolute plane studying
the value of the angle δ defined in any Lambert–Saccheri quadrangle or,
equivalently, the sum of the angles of any triangle. This yelds, in particular, a
Archimedes-free proof of a statement generalizing the classical “first Legendre
theorem” for absolute planes
Three-reflection theorems in the hyperbolic plane
No full textIn this paper we study a generalization of the classical non-Euclidean hyperbolic geometry, without assuming for the absolute plane any condition about continuity or the Archimedes' axiom. In this general frame we extend the validity of the fundamental Three-reflection Theorems to the case of any three distinct lines which are pairwise hyperbolic parallel and have a transversal
Involution sets, graphs with parallelism and loops
No full textThis 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 Sections 4, 5 and 6. In Section 4 we derive a partial binary operation from a bipartite involution set and we discuss if such operation is a Bol operation or a K-operation, in Section 5, Section 6 we relate involution sets with loops
Polar graphs and corresponding involution sets, loops and Steiner triple systems
No full textA 1-factorization (or parallelism) of the complete graph with loops
(P, E, ) is called polar if each 1-factor (parallel class) contains exactly one
loop and for any three distinct vertices x1, x2, x3, if {x1} and {x2, x3} belong
to a 1-factor then the same holds for any permutation of the set {1, 2, 3}.
To a polar graph (P, E,|| ) there corresponds a polar involution set (P, I), an
idempotent totally symmetric quasigroup (P, ∗), a commutative, weak inverse
property loop (P,+) of exponent 3 and a Steiner triple system (P, B).
We have: (P, E,|| ) satisfies the trapezium axiom ⇔ ∀α ∈ I : αIα =
I ⇔(P, ∗) is self-distributive ⇔ (P,+) is a Moufang loop ⇔ (P, B) is an affine
triple system; and: (P, E,|| ) satisfies the quadrangle axiom⇔ I3 =I ⇔(P, +)
is a group ⇔ (P, B) is an affine space
A class of fibered loops related to general hyperbolic planes
No full textIn 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
Loops, reflection structures and graphs with parallelism
No full textThe 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
K-loops derived from Frobenius groups
No full textWe 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 x 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)
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