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Left loops, bipartite graphs with parallelism and bipartite involution sets

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Abstract

We describe a representation of any semiregularleft loop by means of asemiregular 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 toinvariant regular bipartite involution sets and, on the other hand, totrapezium 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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Authors and Affiliations

  1. Zentrum Mathematik, T.U. München, 80290, München, Germany

    H. Karzel

  2. Dipartimento di Matematica e Fisica, Università Cattolica, Via Trieste, 17, 25121, Brescia, Italy

    S. Pianta

Authors
  1. H. Karzel
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  2. S. Pianta
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Correspondence to S. Pianta.

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A. Kreuzer

Research partially supported by the Research Project of M.I.U.R. (Italian Ministry of Education, University and Research) “Strutture geometriche, combinatoria e loro applicazioni” and by the Research group G.N.S.A.G.A. of INDAM.

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Karzel, H., Pianta, S. Left loops, bipartite graphs with parallelism and bipartite involution sets. Abh.Math.Semin.Univ.Hambg. 75, 203–214 (2005). https://doi.org/10.1007/BF02942043

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