  With this chapter we provide a compact yet complete survey of two most
remarkable "representation theorems": every arguesian projective geometry is
represented by an essentially unique vector space, and every arguesian Hilbert
geometry is represented by an essentially unique generalized Hilbert space. C.
Piron's original representation theorem for propositional systems is then a
corollary: it says that every irreducible, complete, atomistic, orthomodular
lattice satisfying the covering law and of rank at least 4 is isomorphic to the
lattice of closed subspaces of an essentially unique generalized Hilbert space.
Piron's theorem combines abstract projective geometry with lattice theory. In
fact, throughout this chapter we present the basic lattice theoretic aspects of
abstract projective geometry: we prove the categorical equivalence of
projective geometries and projective lattices, and the triple categorical
equivalence of Hilbert geometries, Hilbert lattices and propositional systems.
