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 -