Loops, reflection structures and graphs with parallelism

Silvia Pianta, Helmut Karzel, Elena Zizioli

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)74-80
Number of pages7
JournalRESULTATE DER MATHEMATIK
Publication statusPublished - 2002

Keywords

  • graph
  • loop
  • reflection structure

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