Loops, reflection structures and graphs with parallelism.
Geometry and algebraic structures are often only different descriptions of one underlying abstract structure. In this paper, a one-to-one correspondence between semiregular right loops \((P,+)\) with property (*): \(a-(a-b)=b\) \(\forall a, b\in P\) and all graphs with parallelism containing regular vertices is established. In this context \(K\)-loops correspond with trapezium graphs, i.e., graphs with parallelism such that each trapezium \((a,b,c,d)\) satisfies additional axioms T1, T2. A trapezium is defined here as a quadruple \((a,b,c,d)\) of vertices such that \(\{a,b\}\), \(\{b,c\}\), \(\{c,d\}\) are edges and \(\{a,b\}\parallel\{c,d\}\). Axiom T1 postulates that for each trapezium also \(\{d,a\}\) is an edge and axiom T2 postulates for two trapeziums \((a,b,c,d)\), \((a',b',c',d')\) if \(\{a,b\}\parallel\{a',b'\}\), \(\{b,c\}\parallel\{b',c'\}\) then also \(\{a,d\}\parallel\{a',d'\}\).
scientific article; zbMATH DE number 1997673