Extending Partial Projective Planes

If a finite partial projective plane can be extended to a projective plane of order n, then it can be extended by a sequence of one line extensions. We describe necessary conditions for such an extension to exist, and restrictions which can be imposed on the intermediate sequence of extensions. This method can be used to attempt to construct a projective plane of non-prime-power order. 1. Partial Projective Planes and Semiplanes There are various notions of what constitutes a partial projective plane. The most general one is that a partial projective plane is a structure Π = 〈P,L,≤〉 where P is a set of points, L is a set of lines, and p ≤ ` means that the point p is on the line `, satisfying the following two (equivalent) axioms. I. Any two points lie on at most one line. II. Any two lines intersect in at most one point. These axioms allow us to treat a partial projective plane as a partial lattice, with partially defined operations ∨ and ∧. Thus p ∨ q denotes the line through p and q, if there is one, and similarly k ∧ ` denotes the point on both k and `, if there is one. In this context, it is best to avoid think of a line as a set of points, as is normally useful. We say that another partial projective plane, Π′ = 〈P ′, L′,≤′〉, is an extension of Π if P ⊆ P ′, L ⊆ L′, and ≤ is the restriction of ≤′ to P × L. This is denoted by Π v Π′. Every partial projective plane can be extended to a projective plane, possibly infinite; see M. Hall [6]. The questions we want to address are: 1. Can every finite partial projective plane be extended to a finite projective plane? 2. Which finite partial projective planes can be extended to a finite projective plane? 3. Which finite partial projective planes can be extended to a projective plane of order n? 4. How can you extend a finite partial projective plane to a projective plane? Since the first two questions appear to be hard, we concentrate on the third and fourth. 1991 Mathematics Subject Classification. 51E14, 51A35.