On homomorphism-homogeneous point-line geometries

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. A point-line geometry is a non-empty set of elements called points, together with a collection of subsets, called lines, in a way that every line contains at least two points and any pair of points is contained in at most one line. A line which contains more than two points is called a regular line. Point-line geometries can alternatively be formalised as relational structures. We establish a correspondence between the point-line geometries investigated in this paper and the firstorder structures with a single ternary relation L satisfying certain axioms (i.e. that the class of point-line geometries corresponds to a subclass of 3-uniform hypergraphs). We characterise the homomorphism-homogeneous point-line geometries with two regular non-intersecting lines. Homomorphism-homogeneous pointline geometries containing two regular intersecting lines have already been classified by Masulovic.