The correspondence between right loops (P, +) with the property “(*) ∀a, b ∈ P : a − (a − b) − b” and reflection structures described in  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.
AMS subject classification
right loops graphs with parallelism
This is a preview of subscription content, log in to check access.