Loops, reflection structures and graphs with parallelism.
type: Item
instance of: scholarly article
title: Loops, reflection structures and graphs with parallelism.
zbMATH Open document ID: 1046.20043
DOI: 10.1007/BF03323555
author: Helmut Karzel
author: Silvia Pianta
author: Elena Zizioli
published in: Results in Mathematics
publication date: 2003-10-29T00:00:00Z
review text: 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 \(\{a,d\}\parallel\{a',d'\}\).
reviewed by: Heinrich Wefelscheid
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 51A15
zbMATH DE Number: 1997673
zbMATH Keywords: right loops
zbMATH Keywords: \(K\)-loops
zbMATH Keywords: graphs with parallelism
MaRDI profile type: Publication
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cites work: Connections between loops of exponent 2, reflection structures and complete graphs with parallelism
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description: scientific article; zbMATH DE number 1997673