Embedding Finite Partial Linear Spaces in Finite Projective Planes - PDF document

Embedding Finite Partial Linear Spaces in Finite Projective Planes G. Eric Moorhouse, University of Wyoming 1

A partial linear space (PLS) is a pair Γ = ( P , L ) consisting of a set P ( points ) and a collection L of distinguished subsets of P (called lines ) such that (i) each line contains at least two points, and (ii) any two distinct lines meet in at most one point. A point-line pair ( P, ℓ ) in Γ is called a flag or an antiflag according as P ∈ ℓ or P / ∈ ℓ . 2

Let Γ = ( P , L ) and � Γ = ( � P , � L ) be two partial linear spaces. An embedding α : Γ → � Γ is a pair of injections α 1 : P → � α 2 : L → � P , L such that for all P ∈ P , ℓ ∈ L , P ∈ ℓ ⇒ α 1 ( P ) ∈ α 2 ( ℓ ) . Such an embedding is strong if P ∈ ℓ ⇐ ⇒ α 1 ( P ) ∈ α 2 ( ℓ ) . 3