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. P-algebras have type ⟨X, 0,^′, ·⟩ where 0 is a constant,^′ is unary and · is binary. Elements of X are called features. A partial order is defined on the set X of features by x ≤ y iff x · y = x. Features commute, i.e., x · y = y · x iff x · y ≤ x. Features x and y are said to be orthogonal iff x · y = 0 and orthogonality is a symmetric relation. The operation + is defined as the dual of · and it is commutative on orthogonal features. 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 but they are not lattices. Existence of least upper bounds for ascending sequences is equivalent to the existence of least upper bounds for countable sets of pairwise orthogonal elements. Atomic algebras are defined and their main properties are studied. The logic of P-algebras is then completely characterized. The language contains a unary connective corresponding to the operation^′ and a binary connective corresponding to the operation “·”. It is a substructural logic of sequents where the Exchange rule is extremely limited. It is proved to be sound and complete for P-algebras.