Projection-algebras and quantum logic

P-algebras are a non-commutative, non-associative generalization of Boolean algebras that are for Quantum Logic what Boolean algebras are for Classical Logic.The closed subspaces of a separable Hilbert space form a P-algebra under orthogonal complementation and projection of a subspace onto another one. P-algebras are complemented orthomodular posets that are not lattices. Atomic algebras are defined and their main properties are studied. A substructural logic of sequents is proved to be sound and complete for the logic of P-algebras.