Unordered love in infinite directed graphs (Q1205616)

The results and problems of this paper concern relations between projective planes and digraphs with the SDUL Property. Digraph \(D=(V,A)\) has: (a) the UL Property, if \(u,v\in V\) and \(u\neq v\) imply the existence of a unique \(w\) such that \((u,w)\), \((v,w)\in A\); (b) the SDUL Property, if both \((V,A)\) and \((V,A^{-1})\) have the UL Property. The author has proven the following results which are known in the case of finite digraphs, for \(D\) infinite: (i) if \(D\) is loopless and has the SDUL Property, then either \(D\) has a vertex \(v\) connected both ways to every other vertex, such that \(D-v\) is a disjoint union of directed cycles or \(D\) is associable with a projective plane, obtainable by taking \(V\) as the set of points and the sets of outneighbours of vertices as the lines; (ii) every projective plane arises from a digraph with the SDUL Property, in this way.