From involution sets, graphs and loops to loop-nearrings. (English) Zbl 1097.20054
Kiechle, Hubert (ed.) et al., Nearrings and nearfields. Proceedings of the conference on nearrings and nearfields, Hamburg, Germany, July 27–August 3, 2003. Dordrecht: Springer (ISBN 1-4020-3390-7/hbk; 1-4020-3391-5/e-book). 235-252 (2005).
The authors in recent papers established connections between \(K\)-loops, invariant reflection structures and certain graphs with parallelism. The notion of an involution set gives rise from one side to a binary operation and from the other side to geometric structure of a graph with parallelism.
In this paper the authors study the connections between these structures and translate properties of involution sets into configurational properties of the associated graphs. They show that the automorphism group of the left loop \((P,+)\) coincides with the group of automorphisms of the involution \((P,I)\), and of the graph, fixing the vertex and they relate such automorphism groups to the construction of loop-nearrings.
For the entire collection see [Zbl 1061.16001].
In this paper the authors study the connections between these structures and translate properties of involution sets into configurational properties of the associated graphs. They show that the automorphism group of the left loop \((P,+)\) coincides with the group of automorphisms of the involution \((P,I)\), and of the graph, fixing the vertex and they relate such automorphism groups to the construction of loop-nearrings.
For the entire collection see [Zbl 1061.16001].
Reviewer: C. Pereira da Silva (Curitiba)
MSC:
| 20N05 | Loops, quasigroups |
| 05C75 | Structural characterization of families of graphs |
| 51A15 | Linear incidence geometric structures with parallelism |
| 16Y30 | Near-rings |
| 20N02 | Sets with a single binary operation (groupoids) |
| 05C15 | Coloring of graphs and hypergraphs |
