Propositional systems, Hilbert lattices and generalized Hilbert spaces

TL;DR

This paper surveys key representation theorems linking projective geometries, Hilbert geometries, and generalized Hilbert spaces, highlighting their categorical equivalences and lattice-theoretic foundations.

Contribution

It provides a comprehensive overview of theorems connecting geometric and algebraic structures, emphasizing the categorical equivalences and lattice representations involved.

Findings

Representation of arguesian projective geometries by vector spaces

Representation of arguesian Hilbert geometries by generalized Hilbert spaces

Categorical equivalences among projective geometries, Hilbert geometries, Hilbert lattices, and propositional systems

Abstract

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.