命题系统、Hilbert格和广义Hilbert空间
Propositional systems, Hilbert lattices and generalized Hilbert spaces
摘要：
在本章中，我们将对两个最显著的“表示定理”进行一个紧凑而完整的考察：每个笛卡尔射影几何都由一个本质上唯一的向量空间表示，每个笛卡尔希尔伯特几何都由一个本质上唯一的广义希尔伯特空间表示。C. Piron的命题系统的原始表示定理是一个推论：它说每个满足覆盖律且秩至少为4的不可约、完备、原子、正交模格同构于本质上唯一的广义希尔伯特空间的闭子空间格。Piron定理将抽象射影几何与格理论相结合。事实上，在这一章中，我们介绍了抽象射影几何的基本格理论方面：我们证明了射影几何和射影格的范畴等价性，以及希尔伯特几何、希尔伯特格和命题系统的三元范畴等价性。
英文摘要：
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.
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