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

Abstract

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.

Our two main results can be read independently of each other. The first utilises Priestley duality to prove that every regular double p-algebra can be embedded into a lattice of point-preserving substructures of an incidence structure. The second main result is a characterisation of the regular double p-algebras which are isomorphic to a lattice of point-preserving substructures. In addition to the corollary that every finite regular double p-algebra is isomorphic to a lattice of point-preserving substructures, a special case of the second result is a standard theorem for boolean algebras: a boolean algebra B is isomorphic to a powerset lattice if and only if it is complete and atomic.