A Characterisation of Orthomodular Spaces by Sasaki Maps

Abstract

Given a Hilbert space H, the set P(H) of one-dimensional subspaces of H becomes an orthoset when equipped with the orthogonality relation ⊥ induced by the inner product on H. Here, an orthoset is a pair (X,⊥) of a set X and a symmetric, irreflexive binary relation ⊥ on X. In this contribution, we investigate what conditions on an orthoset (X,⊥) are sufficient to conclude that the orthoset is isomorphic to (P(H),⊥) for some orthomodular space H, where orthomodular spaces are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce Sasaki maps on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset (X,⊥) with sufficiently many Sasaki maps is isomorphic to (P(H),⊥) for some orthomodular space, and we give more conditions on (X,⊥) to assure that H is actually a Hilbert space over $\mathbb R$ ℝ , $\mathbb C$ ℂ or $\mathbb H$ ℍ .