Projective lines as groupoids with projection structure
The author defines a projection structure on a groupoid \(\mathbb{L}\) as a set of bijections between the set of arrows \(A \to B\) and \(\mathbb{L}\setminus\{A,B\}\) for any two distinct points \(A\), \(B \in \mathbb{L}\). The projective line \(P(V)\) over a two-dimensional \(k\)-vector space \(V = k^2\) is naturally equipped with such a projection structure: The restriction of the vector space isomorphism taking the subspace \(A\) to the subspace \(B\) is a projection parallel to subspace \(C \neq A, B\). The aim of this article is to recover the vector space from an axiomatically defined projection structure. To this end, the author introduces four axioms: Axioms 1 and 2 concern commutativity properties of arrows, Axioms 3 and 4 are concern permutation properties of bi-rapports (cross-ratios) and tri-rapports.NEWLINENEWLINEThe scalars are defined as elements of the abstract vertex group (all vertex groups are isomorphic). Hence, bi- and tri-rapports are compositions of certain arrows. The author relates bi- and tri-rapports, develops some basic arithmetic properties of the scalar group, proves a fundamental theorem and, under some mild additional assumptions that make the scalar group a field, identify the groupoid with the projective line over its scalar field.NEWLINENEWLINEThe article closes with the suggestion to develop a weaker axiomatic theory that also covers projective lines over rings.
scientific article; zbMATH DE number 6334031