Algebras of incidence structures: representations of regular double p-algebras

An incidence structure is a standard geometric object consisting of a set of points, a set of lines and an incidence relation specifying which points lie on which lines. This concept generalises, for example, both graphs and projective planes. We prove that the lattice of point-preserving substructures of an incidence structure naturally forms a regular double p-algebra. A double p-algebra A is regular if for all \({x, y \,\in \, A}\), we have that x⁺ =  y⁺ and x* =  y* together imply x = y.