We describe a representation of any semiregular left loop by means of a semiregular bipartite involution set or, equivalently, a 1-factorization (i.e., a parallelism) of a bipartite graph, with at least one transitive vertex. In these correspondences, Bol loops are associated on one hand to invariant regular bipartite involution sets and, on the other hand, to trapezium complete bipartite graphs with parallelism; K-loops (or Bruck loops) are further characterized by a sort of local Pascal configuration in the related graph.
Karzel, H., Pianta, S., Left loops, bipartite graphs with parallelism and bipartite involution sets, <<ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITAT HAMBURG>>, 2005; (75): 203-214. [doi:10.1007/BF02942043] [http://hdl.handle.net/10807/55453]
Left loops, bipartite graphs with parallelism and bipartite involution sets
Karzel, Helmut;Pianta, Silvia
2005
Abstract
We describe a representation of any semiregular left loop by means of a semiregular bipartite involution set or, equivalently, a 1-factorization (i.e., a parallelism) of a bipartite graph, with at least one transitive vertex. In these correspondences, Bol loops are associated on one hand to invariant regular bipartite involution sets and, on the other hand, to trapezium complete bipartite graphs with parallelism; K-loops (or Bruck loops) are further characterized by a sort of local Pascal configuration in the related graph.
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