Distributivity and perspectivity in orthomodular lattices
Samuel S. Holland · Transactions of the American Mathematical Society · 1964
1. Introduction.In this paper we prove for complete orthomodular lattices analogues of two results of continuous geometry.By (a, b)P, where a, b are lattice elements von Neumann means that aAb = 0 and that for any x in the lattice all six possible distributive laws for the triple (a, b, x) hold [13, Part I, proof of Theorem 5.8].The first result we consider is von Neumann's theorem: in a continuous geometry (not necessarily irreducible) if iax,b) P holds for all a in some indexing set A, then, setting a = f\iax;<x,eA), (a, b)P holds also [13, Part I, Theorem 5.8].Kaplansky observed that this theorem is true in any complete complemented modular lattice, continuous or not [5, Lemma 36].In this paper we prove that von Neumann's theorem holds in a complete orthomodular lattice, assuming neither modularity nor continuity (Theorem 1).The second result we consider is Lemma 29 of [5].In [5] Kaplansky proves: if iax;aeA), ibx;<xeA) are two families of elements from a complete orthocomplemented modular lattice indexed by the same set A such that ax and bx are perspective for every a in A ("perspective" means they share a common complement) and such that ax V bx is orthogonal to aß V bß for a # ß, then V ax and f\ bx are also perspective.We show (Theorem 3) in Kaplansky's result "orthocomplemented modular" can be replaced by "orthomodular," thus generalizing the result to a larger class of lattices.However to achieve the generalization we need to use the following form of the definition of the perspectivity of a and b : that there exists x such that a\jx = b\/x = a\Jb, a f\x = b Ax = 0.In a complemented modular lattice this is equivalent to requiring that a and b have a common complement [13, Part I, Theorem 3.1].In an orthomodular lattice the two conditions are equivalent if and only if the lattice is modular (Theorem 2).Orthomodular lattices derive their principal interest from the fact that they