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.

翻译：P-代数是布尔代数的一种非交换、非结合推广，其在量子逻辑中的地位相当于布尔代数在经典逻辑中的地位。可分离希尔伯特空间的闭子空间在正交补及子空间到另一子空间的投影运算下构成一个P-代数。P-代数是一种具有补的弱正交模偏序集，而非格结构。本文定义了原子代数并研究了其主要性质。一种关于相继式的子结构逻辑被证明对于P-代数的逻辑具有可靠性和完备性。