TY - JOUR

T1 - Loops, reflection structures and graphs with parallelism

AU - Pianta, Silvia

AU - Karzel, Helmut

AU - Zizioli, Elena

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - graph

KW - loop

KW - reflection structure

KW - graph

KW - loop

KW - reflection structure

UR - http://hdl.handle.net/10807/55488

U2 - 10.1007/BF03323555

DO - 10.1007/BF03323555

M3 - Article

SN - 0378-6218

VL - 42

SP - 74

EP - 80

JO - RESULTATE DER MATHEMATIK

JF - RESULTATE DER MATHEMATIK

ER -