Projective rectangles 

This talk introduces the concept of a projective rectangle (PR), a generalization of the projective plane. This incidence structure comprises points and lines, with a distinction between special lines, each containing a unique special point, and ordinary lines. In a projective plane---a familiar example of a PR---any point can serve as the special point, particularly in constructions over finite fields. However, in more general settings, there exists a single unique special point.

We present various examples of projective rectangles and examine their connections to full algebraic matroids, specifically harmonic matroids. Furthermore, we investigate their relationships with graph theory, biased graphs, and orthogonal arrays. This ongoing work, conducted in collaboration with T. Zaslavsky, highlights the current progress and key findings of our research.